Download TANGENTS AND SECANTS OF ALGEBRAIC VARIETIES F. L. Zak

TANGENTS AND SECANTS OF ALGEBRAIC VARIETIES
F. L. Zak
Central Economics Mathematical Institute
of the Russian Academy of Sciences
CONTENTS
Index of Notations
Introduction
Chapter I. Theorem on Tangencies and Gauss Maps
1 Theorem on tangencies and its applications
2 Gauss maps of projective varieties
3 Subvarieties of complex tori
Chapter II. Projections of Algebraic Varieties
1 A criterion for existence of good projections
2 Hartshorne’s conjecture on linear normality and its relative
analogues
Chapter III. Varieties of Small Codimension Corresponding to Orbits of Algebraic Groups
1 Orbits of algebraic groups, null-forms and secant varieties
2 HV -varieties of small codimension
3 HV -varieties as birational images of projective spaces
Chapter IV. Severi Varieties
1 Reduction to the nonsingular case
2 Quadrics on Severi varieties
3 Dimension of Severi varieties
4 Classification theorems
5 Varieties of codegree three
Chapter V. Linear Systems of Hyperplane Sections on
Varieties of Small Codimension
1 Higher secant varieties
2 Maximal embeddings of varieties of small codimension
Chapter VI. Scorza Varieties
1 Properties of Scorza varieties
2 Scorza varieties with δ = 1
3 Scorza varieties with δ = 2
4 Scorza varieties with δ = 4
5 End of classification of Scorza varieties
Bibliography
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CHAPTER I
THEOREM ON TANGENCIES AND GAUSS MAPS
Typeset by AMS-TEX
14
1. THEOREM ON TANGENCIES AND ITS APPLICATIONS
15
1. Theorem on tangencies and its applications
Let X n ⊂ PN be an irreducible nondegenerate (i.e. not contained in a hyperplane) n-dimensional projective variety over an algebraically closed field K, and let
Y r ⊂ PN be a non-empty irreducible r-dimensional variety. We set
¯
©
ª
∆Y = (Y × X) ∩ ∆X = (y, x) ∈ Y × X ¯ x = y ,
where ∆X is the diagonal in X × X,
0
SY,X
⊂ (Y × X \ ∆Y ) × PN ,
¯
©
ª
0
SY,X
= (y, x, z) ¯ z ∈ hx, yi ,
where hx, yi denotes the chord joining x with y. We denote by SY,X the closure of
0
SY,X
in Y ×X ×PN , by pYi the projection of SY,X onto the ith factor of Y ×X ×PN
(i = 1, 2), and by ϕY : SY,X → PN the projection onto the third factor, and put
pY12 = pY1 × pY2 : SY,X → Y × X,
¯
¡ ¢−1
0
= pY12
(∆Y ) ,
ψ Y = ϕ Y ¯T 0
TY,X
Y,X
,
S(Y, X) = ϕY (SY,X ),
¡ 0 ¢
T 0 (Y, X) = ψ Y TY,X
.
1.1. Definition. The variety S(Y, X) is called the join of Y and X or, if
Y ⊂ X, the secant variety of X with respect to Y .
We observe that in the case when Y = X the above definition reduces to the
usual definition of secant variety of X; we shall denote S(X, X) simply by SX.
In what follows we shall assume that Y r ⊂ X n is a subvariety of X.
1.2. Definition. The variety T 0 (Y, X) is called the variety of (relative) tangent
stars of X with respect to the subvariety Y .
We observe that T 0 (X, X) = T 0 X is the usual variety of tangent stars (cf. [45;
97]).
´
³¡ ¢
−1
0
(y × y) is called the (pro1.3. Definition. The cone TY,X,y
= ψ Y pY12
jective) tangent star to X with respect to Y ⊂ X at a point y ∈ Y .
0
From this definition it is evident that TY,X,y
is a union of limits of chords
h y 0 , x0 i,
y 0 ∈ Y, x0 ∈ X,
y 0 , x0 → y.
0
0
0
0
It is also clear that TY,X,y
⊂ TX,y
⊂ TX,y , where TX,y
= TX,X,y
is the (projective)
tangent star to X at y (cf. [45; 97]) and TX,y is the (embedded) tangent space to
0
0
0
0
X at y. On the other hand, TY,X,y
⊃ Ty,X
, where Ty,X
= Ty,X,y
is the (projective)
tangent cone to X at the point y.
0
By definition, T 0 (Y, X) = ∪ TY,X,y
. If X is nonsingular along Y , i.e. Y ∩
y∈Y
Sing X = ∅ and Y ⊂ Sm X = X \ Sing X, then T 0 (Y, X) = T (Y, X) = ∪ TX,y is
y∈Y
the usual variety of tangents.
16
I. THEOREM ON TANGENCIES AND GAUSS MAPS
1.4. Theorem. An arbitrary irreducible subvariety Y r ⊂ X n , r ≥ 0 satisfies
one of the following two conditions:
a) dim T 0 (Y, X) = r + n,
b) T 0 (Y, X) = S(Y, X).
dim S(Y, X) = r + n + 1;
Proof. Let t = dim T 0 (Y, X). It is clear that t ≤ r +n. In the case when t = r +n
the theorem is obvious since S(Y, X) is an irreducible variety, S(Y, X) ⊃ T 0 (Y, X)
and dim S(Y, X) ≤ r + n + 1.
Suppose that t < r + n, and let LN −t−1 be a linear subspace of PN such that
L ∩ T 0 (Y, X) = ∅.
(1.4.1)
We denote by π: PN ¯\ L → Pt the projection with center at L and put X 0 = π(X),
Y 0 = π(Y ). Since π¯X is a finite morphism, we have dim (Y 0 × X 0 ) = r + n > t,
and from the connectedness theorem of Fulton and Hansen (cf. [26] and [27, 3.1])
it follows that
¯ ¢−1
¡ ¯
Y ×X 0 X = π¯Y ×π ¯X
(∆Pt )
is a connected scheme.
I claim that
Supp (Y ×X 0 X) = ∆Y .
(1.4.2)
In fact, suppose that this is not so. Then by definition for all (y, x) ∈ (Y ×X 0 X)\∆Y
we have
³¡ ¢−1
´
ϕY pY12
(y, x) ∩ L 6= ∅,
and therefore for each point (y, y) ∈ ∆Y ∩ ((Y ×X 0 X) \ ∆Y )
0
T 0 (Y, X) ∩ L ⊃ TY,X,y
∩ L = ϕY
³¡ ¢−1
´
pY12
(y, y) ∩ L 6= ∅
contrary to (1.4.1). This proves (1.4.2).
From (1.4.2) it follows that L ∩ S(Y, X) = ∅. Hence
t ≤ dim S(Y, X) ≤ N − dim L − 1 = t,
i.e. condition b) holds. ¤
1.5. Corollary.
codimS(Y,X) T 0 (Y, X) ≤ 1.
1.6. Definition. Let L ⊂ PN be a linear subspace. We say that L is tangent
to a variety X ⊂ PN along a subvariety Y ⊂ X (resp. L is J-tangent to X along
0
Y , resp. L is J-tangent to X with respect to Y ) if L ⊃ TX,y (resp. L ⊃ TX,y
, resp.
0
L ⊃ TY,X,y ) for all points y ∈ Y .
It is clear that if L is tangent to X along Y , then L is J-tangent to X along Y
and if L is J-tangent to X along Y , then L is J-tangent to X with respect to Y .
If X is nonsingular along Y , then all the three notions are identical.
1. THEOREM ON TANGENCIES AND ITS APPLICATIONS
17
1.7. Theorem. Let Y r ⊂ X n and Z b ⊂ Y r be closed subvarieties, and let
L ⊂ PN , n ≤ m ≤ N −1 be a linear subspace which is J-tangent to X with respect
0
to Y along Y \ Z (i.e. L ⊃ TY,X,y
for all points y ∈ Y \ Z). Then r ≤ m − n + b + 1.
m
Proof. It is clear that Theorem 1.7 is true (and meaningless) for r ≤ b + 1.
Suppose that r > b + 1. Without loss of generality we may assume that Y is
irreducible. Let M be a general linear subspace of codimension b + 1 in PN . Put
X 0 = X ∩ M,
It is clear that
Y 0 = Y ∩ M,
L0 = L ∩ M.
n0 = dim X 0 = n − b − 1,
r0 = dim Y 0 = r − b − 1,
0
(1.7.1)
0
m = dim L = m − b − 1
and L0 is J-tangent to X 0 with respect to Y 0 along Y 0 . In other words,
T 0 (Y 0 , X 0 ) ⊂ L0 .
(1.7.2)
In particular, from (1.7.2) it follows that
dim T 0 (Y 0 , X 0 ) ≤ m0 .
(1.7.3)
Since n > r > b+1, from the Bertini theorem it follows that the varieties X 0 and Y 0
are irreducible. By [58, Lema 1, Corolario 1], the variety X 0 is nondegenerate, and
so the relative secant variety S(Y 0 , X 0 ) containing X 0 does not lie in the subspace
L0 . From (1.7.2) it follows that
S(Y 0 , X 0 ) 6= T 0 (Y 0 , X 0 ).
(1.7.4)
In view of (1.7.4) Theorem 1.4 yields
dim T 0 (Y 0 , X 0 ) = r0 + n0 .
(1.7.5)
Combining (1.7.3) and (1.7.5) we see that r0 + n0 ≤ m0 , and in view of (1.7.1)
r ≤ m − n + b + 1. ¤
1.8. Corollary (Theorem on tangencies). If a linear subspace Lm ⊂ PN is
tangent to a nondegenerate variety X n ⊂ PN along a closed subvariety Y r ⊂ X n ,
then r ≤ m − n.
1.9. Remark. It is clear that if Z does not contain components of Y , then in
the statement of Theorem 1.7 we may assume that Z ⊂ Y ∩ Sing X.
We give an example showing that the bound in Theorem 1.7 is sharp.
1.10. Example. Let X n ⊂ PN , N = 2n − b − 2 be a cone with vertex Pb over
the Segre variety P1 × Pn−b−2 ⊂ P2n−2b−3 , n > b + 2. Then
X ∗ = (X 0 )∗ = P1 × Pn−b−2 ⊂ (Pb )∗ = P2n−2b−3 ,
18
I. THEOREM ON TANGENCIES AND GAUSS MAPS
and a subspace Lm ⊂ PN , n ≤ m ≤ N − 1 is tangent to X at a point x ∈ Sm X
(and all points of the (b + 1)-dimensional affine linear space hx, Pb i \ Pb ) if and only
if the (N − m − 1)-dimensional linear subspace L∗ is contained in the (N − n − 1)∗
dimensional linear subspace TX,x
⊂ X ∗ (here and in what follows asterisk denotes
dual variety and hAi denotes the linear span of a subset A ⊂ PN ). It is easy to
see that an arbitrary (n − b − 3)-dimensional linear subspace lying in X ∗ coincides
∗
with TX,x
for some x ∈ X.
Let
Pn−b−2 ⊂ X ∗ = P1 × Pn−b−2
be a linear subspace, and let L∗ be an arbitrary (N − m − 1)-dimensional linear
subspace of Pn−b−2 . Then the m-dimensional linear subspace L = (L∗ )∗ is tangent
to X at all points of Y \ Pb , where
¯
ª
©
∗
Y = Pm−n+b+1 ⊃ Pb ,
⊂ Pn−b−2 .
Y = x ∈ X ¯ L∗ ⊂ TX,x
Thus for the subspace L and the subvarieties Y = Pm−n+b+1 ⊂ Pn−1 ⊂ X and
Z = Sing X = Pb the inequality in Theorem 1.7 turns into equality.
1.11. Proposition. Let X n ⊂ PN be a nondegenerate variety satisfying
condition Rk (cf. [30, Chapter IV2 , (5.8.2)]) (in other words, X is regular in
codimension k, i.e. b = dim (Sing X) < n − k), and let L be an m-dimensional
linear subspace of PN . Put X 0 = X · L, and let b0 = dim (Sing X 0 ). Then
b0 ≤ 2N − m − n + b − 1 = b + c + ε − 1, i.e. X 0 satisfies condition Rk−c−2ε+1 , where
c = codimPN X = N − n, ε = codimPN L = N − m.
Proof. For an arbitrary point λ of the (ε − 1)-dimensional linear subspace L∗ ⊂
P we put Xλ = X · λ∗ , where λ∗ is the hyperplane corresponding to λ. It is clear
that X 0 = ∩ ∗ Xλ . Let Y = Sing X 0 , Yλ = Sing Xλ , λ ∈ L∗ . It is easy to see that
N∗
λ∈L
Y ⊂ ∪ ∗ Yλ , so that
λ∈L
b0 = dim Y ≤ max∗ bλ + ε − 1,
λ∈L
(1.11.1)
where bλ = dim Yλ . It is clear that the hyperplane λ∗ is tangent to X at all points
of Yλ \ Sing X. Hence from Theorem 1.7 it follows that
bλ ≤ b + c.
(1.11.2)
Combining (1.11.1) and (1.11.2) we obtain the desired bound for b0 . ¤
The following simple example shows that the bound in Proposition 1.11 is sharp.
N −1
1.12.
⊂ PN be a quadratic cone with vertex Pb , and
£ N +b ¤ Example. Let X
let
+ 1 ≤ m ≤ N − 1 (here and in what follows [a] is the largest integer
2
not exceeding a given number a ∈ R). Then X ∗ is a nonsingular quadric in the
(N − b − 1)-dimensional linear subspace (Pb )∗ ⊂ PN ∗ . It is well known (cf. [28,
∗
Volume II, £Chapter
¤ 6; 37, ∗Chapter XIII]) that X contains a linear subspace of
N −b−2
dimension
. Let L be its linear subspace of dimension N − m − 1. Put
2
L = (L∗ )∗ , X 0 = X · L. Then dim L = m, and it is easy to see that Y = Sing X 0 is
an (N − m + b)-dimensional linear subspace.
1. THEOREM ON TANGENCIES AND ITS APPLICATIONS
19
1.13. Corollary. Suppose that a variety X n ⊂ PN satisfies conditions Sε+1 =
SN −m+1 and Rc+2ε−1 = R3N −2m−n−1 , and let Lm ⊂ PN be a linear subspace for
which dim (X · L) = n − ε = m + n − N . Then the scheme X · L is reduced. In
particular, if X is nonsingular, N < 23 (m + n + 1), and dim X · L = m + n − N ,
then X · L is a reduced scheme.
Proof. From Proposition 1.11 it follows that in the conditions of Corollary 1.13
X 0 = X · L satisfies condition R0 . Since dim X 0 = n − ε, X 0 satisfies condition S1
(cf. [61,§ 17]). Hence to prove Corollary 1.13 it suffices to apply Proposition 5.8.5
from [30, Chapter IV2 ]. ¤
1.14. Corollary. If X n ⊂ PN satisfies conditions Sε+2 = SN −m+2 and Rc+2ε =
R3N −2m−n and Lm ⊂ PN is a linear subspace such that dim (X n · Lm ) = n − ε =
m+n−N , then the scheme X ·L is normal (and therefore irreducible and reduced).
In particular, if X is nonsingular, N ≤ 32 (m + n) and dim (X · L) = m + n − N ,
then X · L is a normal scheme.
Proof. From Proposition 1.11 it follows that in the conditions of Corollary 1.14
X 0 = X · L satisfies condition R1 . Since dim X 0 = n − ε, X 0 satisfies condition S2
(cf. [61,§ 17]). Hence to prove Corollary 1.14 it suffices to apply Serre’s normality
criterion (cf. [30, Chapter IV2 , (5.8.6)]). ¤
Of special importance to applications is the case when L is a hyperplane. We
formulate our results in this case.
1.15. Corollary. a) If a variety X n ⊂ PN is nondegenerate and normal and
N ≤ 2n − b − 1, where b = dim (Sing X), then all hyperplane section of
X are reduced. In particular, if X is nonsingular and N < 2n, then all
hyperplane sections of X are reduced.
b) If a nondegenerate variety X n ⊂ PN has properties S3 and RN −n+2 (the
last assumption means that N < 2n − b − 2), then all hyperplane sections
of X are normal (and therefore irreducible and reduced). In particular, if
X is nonsingular and N < 2n − 1, then all hyperplane sections of X are
normal.
1.16. Remark. Corollary 1.15 gives a much more precise information than
Bertini type theorems describing properties of generic hyperplane sections (cf. e.g.
[80]), but, as shown by Examples 1.18 and 1.19 below, the assumptions in its statement cannot be weakened.
1.17. Remark. If K = C and b = −1, then in the assumptions of Corollary 1.15 b) irreducibility of hyperplane sections follows from the Barth-Larsen theorem according to which for N < 2n − 1 the Picard group Pic X ' Z is generated
by the class of hyperplane section of X (cf. [54; 60; 65]).
We give examples showing that the bounds in Corollary 1.15 are sharp.
1.18. Example. Let X0 = P1 × Pn−b−1 ⊂ P2n−2b−1 , n > b + 1, and let
Y0 = x × Pn−b−2 ⊂ X0 be a linear subspace. We denote by X 0 ⊂ P2(n−b−1)
the section of X0 by a general hyperplane passing through Y0 . It is easy to see
that X 0 is a nonsingular projectively normal variety (cf. e.g. [73]). Let X n ⊂ PN
20
I. THEOREM ON TANGENCIES AND GAUSS MAPS
, N = 2n − b − 1 be the projective cone with vertex Pb over X 0 . It is clear that
X is a normal variety and dim (Sing X) = b, so that X satisfies conditions S2 and
Rn−b−1 = RN −n . However X has a non-reduced hyperplane section corresponding
to the hyperplane in P2n−2b−1 which is tangent to X0 along Y0 (cf. Example 1.10).
1.19. Example. Let X0 = Pn−b−2 ⊂ P2n−2b−3 , n > b + 2, and let X be the
projective cone with vertex Pb over X0 . Then X n ⊂ PN , N = 2n − b − 2 is a
Cohen-Macaulay variety (cf. e.g. [47; 73]) and dim (Sing X) = b, so that X satisfies
conditions S3 and Rn−b−1 = RN −n+1 . However for each hyperplane L such that
L∗ ∈ X ∗ = X0∗ L · X is a reducible and therefore non-normal variety, viz. L · X =
H1 ∪ H2 , where H1 = Pn−1 and H2 is the cone with vertex Pb over P1 × Pn−b−3 , is
a reducible and therefore non-normal variety, and Sing (L · X) = H1 ∩ H2 = Pn−2
(cf. Example 1.10).
2. GAUSS MAPS OF PROJECTIVE VARIETIES
21
2. Gauss maps of projective varieties
Let X n ⊂ PN be an irreducible nondegenerate variety. For n ≤ m ≤ N − 1 we
put
¯
©
ª
Pm = (x, α) ∈ Sm X × G(N, m) ¯ Lα ⊃ TX,x ,
where G(N, m) is the Grassmann variety of m-dimensional linear subspaces in PN ,
Lα is the linear subspace corresponding to a point α ∈ G(N, m), and the bar denotes
closure in X × G(N, m). We denote by pm : Pm → X (resp. γm : Pm → G(N, m))
the projection map to the first (resp. second) factor.
2.1. Definition. The map γm is called the mth Gauss map, and its image
∗
= γm (Pm ) is called the variety of m-dimensional tangent subspaces to the
Xm
variety X.
2.2. Remark. Of special interest are the two extreme cases, viz. m = n and
m = N − 1. For m = n we get the ordinary Gauss map γ : X 99K G(N, n), and
∗
for m = N − 1 we see that XN
= X ∗ ⊂ PN is the dual variety and if X is
¡ −1
¢
N −1
nonsingular, then PN −1 = P NPN /X n (−1) , where NPN /X n is the normal bundle
to X in PN (cf. [16, Exposé XVII]).
2.3.
a)
a0 )
b)
b0 )
c)
Theorem. Let dim (Sing X) = b ≥ −1. Then
¡
¢
−1
for each point α ∈ γm p−1
m (Sm X) , dim γm (α) ≤ m − n + b + 1;
∗
dim Xm ≥ (m − n)(N − m − 2) + (m − b − 1); ©
ª
∗
−1
for a general point
α
∈
X
,
dim
γ
(α)
≤
max
b
+
1,
m
+
n
−
N
−
1
m
m
©
ª;
∗
dim Xm ≥ min (m − n)(N − m) + n − b − 1, (m − n + 1)(N − m) + 1 ;
if char K = 0 and γm = νm ◦γ̃m is the Stein factorization of the morphism γm , then νm is a birational isomorphism and the generic fiber of the
morphism γm (and γ̃m ) is a linear subspace of PN of dimension dim Pm −
∗
dim Xm
.
Proof. a) immediately follows from Theorem 1.7, and since
dim Pm = dim X + dim G(N − n − 1, m − n − 1)
= n + (m − n)(N − m),
(2.3.1)
a0 ) follows from a).
−1
b) Suppose first that m = N − 1. It is clear that dim γN
−1 (α) ≤ n − 1, and it
suffices to verify that if n − 1 ≥ b + 2, i.e. n ≥ b + 3, then for a general point α ∈ X ∗
−1
we have dim γN
−1 (α) 6= n − 1. Suppose that this is not so, and let x be a general
point of X. Since n − 1 > b + 1, from Theorem 1.7 it follows that the system of
divisors
¡ −1
¢
∗
Yα = pN −1 γN
α ∈ TX,x
−1 (α) ,
is not fixed, and therefore X = ∪Yα , where α runs through the set of general points
α
∗
∗
of TX,x
. Hence for a general point y ∈ X there exists a hyperplane Λy ⊂ TX,x
such
that for a general point β ∈ Λy we have Lβ ⊃ TX,y . But then
∗
h TX,x , TX,y i ⊂ (Λy ) = Pn+1 ,
22
I. THEOREM ON TANGENCIES AND GAUSS MAPS
i.e. for a general pair of points x, y ∈ X we have
dim (TX,x ∩ TX,y ) = n − 1.
From this it follows that either all n-dimensional linear subspaces from γn (X) are
contained in an (n + 1)-dimensional linear subspace Pn+1 ⊂ PN or they all pass
through an (n − 1)-dimensional subspace Pn−1 ⊂ PN . But in the first case X
is a hypersurface and by Theorem 1.7 dim Yα = n − 1 ≤ b + 1, contrary to our
assumption, and in the second case the intersection of X with a general linear
subspace PN −n+1 ⊂ PN is a nonsingular strange curve (we recall that a projective
curve of degree ≥ 2 is called strange if all its tangent lines pass through a fixed
point). It is well known (cf. [59; 34, ChapterIV; 39 or 75]) that the only nonsingular
strange curves are conics in characteristic 2. Therefore in the second case X is
a quadric, and we again come to a contradiction. Thus assertion b) holds for
m = N − 1 (if char K = 0, then one can simplify the proof using the reflexivity
theorem according to which (X ∗ )∗ = X (cf. [96])).
Next we prove assertion b) for m = k under the assumption that it holds for
∗
m = k + 1. It is clear that for general points αk ∈ Xk∗ , αk+1 ∈ Xk+1
we have
dim Yαk ≤ dim Yαk+1 .
(2.3.2)
If b + 1 ≥ k + n − N , then from the induction hypothesis it follows that
dim Yαk ≤ dim Yαk+1 ≤ b + 1.
Suppose that
dim Yαk+1 ≤ k + n − N > b + 1.
(2.3.3)
If dim Yαk < dim Yαk+1 , then assertion b) for m = k immediately follows from
(2.3.3). Otherwise from (2.3.2) and (2.3.3) it follows that for a general point x ∈ X
∗
and a general point αk+1 ∈ Xk+1
for which Yαk+1 3 x each hyperplane in Lαk+1
containing TX,x is tangent to X at all points of a (dim Yαk+1 )-dimensional component of Yαk+1 that are nonsingular on X, and by Theorem 1.7 dim Yαk+1 ≤ b + 1.
But then
dim Yαk = dim Yαk+1 ≤ b + 1,
so that inequality b) holds also in this case. Assertion b) is proved.
b0 ) immediately follows from b) in view of (2.3.1).
∗
N
c) Let αm be a general point of Xm
. The linear subspace Lm
is tangent
αm ⊂ P
to X at all points of the subvariety
Yαm ∩ Sm X,
¡ −1
¢
Yαm = pm γm
(αm ) ,
and it is easy to see that
Yαm ∩ Sm X =
\
Lα ⊃Lαm
¡
¢
Yα ∩ Sm X ,
(2.3.4)
2. GAUSS MAPS OF PROJECTIVE VARIETIES
23
where α runs through the set of points of X ∗ for which Lα ⊃ Lαm . From the
reflexivity theorem (cf. e.g. [49]) it follows that if char K = 0, then for a general
point α ∈ X ∗ we have
¡ −1
¢
∗
Yα = pN −1 γN
(2.3.5)
−1 (α) = (TX ∗ ,α )
is a linear subspace of PN of dimension N − dim X ∗ − 1. From (2.3.4) and (2.3.5)
it follows that
\
∗
Yαm = Yαm ∩ Sm X =
(TX ∗ ,α )
Lα ⊃Lαm
is also a linear subspace of PN . Since char K = 0, the morphism γm is separable
and therefore smooth at a general point. Hence νm is a birational isomorphism.
This completes the proof of assertion c) and Theorem 2.3. ¤
We observe that if char K = p > 0, then assertion c) of Theorem 2.3 is no longer
true. As an example, it suffices to consider the hypersurface in Pn+1 defined by
n+1
P p+1
equation
xi
= 0 (in this case γ is the Frobenius map). The case of positive
i=0
characteristic is treated in [50].
2.4. Corollary. If char K = 0, X n ⊂ PN is a nonsingular variety, and N −
n + 1 ≤ m ≤ N − 1, then a general m-dimensional tangent subspace is tangent to
X along a linear subspace of dimension at most m + n − N − 1 (for N ≥ 2n this
bound is better than the one given in Theorem 1.7). For n ≤ m ≤ N − n + 1 a
general m-dimensional tangent subspace is tangent to X at a single point.
2.5. Corollary. Let X n ⊂ PN , X n 6= Pn , n∗ = dim X ∗ , b = dim (Sing X).
Then n∗ ≥ n − b − 1. In particular, for a nonsingular variety n∗ ≥ n. If n ≥ b + 3,
then n ≥ N − n + 1 (this bound is better than the preceding one if N ≥ 2n − b − 1).
The following example shows that both bounds in Corollary 2.5 are sharp.
2.6. Example. Let X0 = P1 × Pn−b−2 ⊂ P2n−2b−3 , n > b + 2, and let X be
a projective cone with vertex Pb and base X0 . Then X n ⊂ PN , N = 2n − b − 2,
dim (Sing X) = b, X ∗ = X0∗ ' X0 , and n∗ = n − b − 1 = N − n + 1.
2.7. Remark. In the case when char K = 0 and b = −1, the inequality n∗ ≥
N − n + 1 was independently proved by Landman (cf. [50]). Another proof was
earlier given by the author (cf. [96, Proposition 1] for n = 2; the general case is
quite similar).
2.8. Corollary. Let X n ⊂ PN , X n 6= Pn , b = dim (Sing X). Then dim γ(X) ≥
n − b − 1. In particular, for a nonsingular variety, dim γ(X) = dim X and γ is a
finite morphism. If in addition char K = 0, then γ is a birational isomorphism (i.e.
γ is the normalization morphism).
2.9. Remark. In the case when K = C and b = −1, Griffiths and Harris [29]
proved that dim γn (X) = dim X. Different proofs of finiteness of γn in this case
were later given by Ein [18] and Ran [68]. In our first proof of Corollary 2.8 (and
Theorem 1.7) we used methods of formal geometry. Since related techniques is used
in §3, we give this proof here.
24
I. THEOREM ON TANGENCIES AND GAUSS MAPS
As in the proof of Theorem 1.7, considering the intersection of X with a general
(N −b−1)-dimensional linear subspace of PN we reduce everything to the case when
b = −1. Suppose that the n-dimensional linear subspace L corresponding to a point
αL ∈ G(N, n) is tangent to X along an irreducible subvariety Y , dim Y > 0, i.e.
Y ⊂ γ −1 (αL ). Let X = X/Y be the completion of X along Y , and let G = γ(X)/αL
be the formal neighborhood of the point αL in the variety γ(X) ⊂ G(N, n). Since
X n 6= Pn , dim γ(X) > 0. Hence H 0 (G, OG ) and H 0 (X, OX ) ⊃ H 0 (G, OG ) are
infinite-dimensional vector spaces over the field K.
On the other hand, let M ⊂ PN be a linear subspace, dim M = N − n − 1,
M ∩ L = ∅, and let π : X 99K Pn be the projection with center at M . Then
π/Y : X → Pn/π(Y ) is an isomorphism of formal spaces, and therefore
H 0 (X, OX ) ' H 0 (L, OL ) ,
(2.9.1)
where L = L/Y ' Pn/π(Y ) is the completion of L along Y . But by the well-known
theorem on formal functions (cf. [31, Chapter V; 36]), H 0 (L, OL ) = K which is
impossible since H 0 (X, OX ) is infinite-dimensional in view of (2.9.1).
The above contradiction shows that dim Y = 0, i.e. γ is a finite morphism.
Although, as we have already seen, the bounds in Theorem 2.3 are sharp, one
can still prove stronger results for certain special classes of projective varieties. An
important example is given by complete intersections.
2.10. Proposition. Let X n ⊂ PN be a nondegenerate nonsingular complete
∗
=
intersection. Then all Gauss maps γm , n ≤ m ≤ N − 1 are finite and dim Xm
dim Pm = n+(m−n)(N −m). If in addition char K = 0, then all γm , n ≤ m ≤ N −1
are birational isomorphisms.
∗
Proof. Let αm ∈ Xm
, α ∈ X ∗ be points for which there is an inclusion of
−1
the corresponding linear subspaces Lαm ⊂ Lα . Then it is clear that γm
(αm ) ⊂
−1
γN −1 (α). Hence it suffices to prove Proposition 2.10 in the case when m = N − 1.
¢
¡
We recall that PN −1 = P NPN /X n (−1) (cf. Remark 2.2). Furthermore, the
∗
morphism γN −1 : PN −1 → XN
−1 is ¯defined by¯ a linear subsystem without fixed
¯OP
points of the complete
linear
system
(1)¯, where OPN −1 (1) is the tautologN −1
¡
¢
ical sheaf on P NPN /X n (−1) (cf. [16, Exposé XVII]). In view of [30, Chapter II,
6.6.3] and [31, Chapter III], to show that γN −1 is finite it suffices to verify that
NPN /X n (−1) is an ample vector bundle. But if X is complete intersection of hypersurfaces Fi ,
deg Fi = ai ≥ 2,
i = 1, . . . , N − 1,
then
N −n
NPN /X n (−1) = ⊕ OX (ai − 1),
i=1
and by [31, Chapter III] NPN /X n (−1) is an ample bundle.
The remaining assertions of Proposition 2.10 follow from (2.3.1) and assertion
c) of Theorem 2.3. ¤
2.11. Remark. The above proof of Proposition 2.10 can also be interpreted in
elementary terms; cf. [42].
2. GAUSS MAPS OF PROJECTIVE VARIETIES
25
The Gauss map γ : X → G(N, n), where X n ⊂ PN , X n 6= Pn is a nonsingular
variety, can also be interpreted in another way. To begin with, γ is the map
corresponding to the vector bundle NPN /X n (−1) with a distinguished (N + 1)dimensional vector subspace of sections corresponding to points of K N +1 (where
PN = (K N +1 \ 0)/K ∗ ; cf. [28]).
Furthermore, let L ⊂ PN , dim L = N −n−1 be a general linear subspace, and let
πL : X → Pn be the projection with center in L. We denote by RL the ramification
divisor of the finite covering πL ,
¯
ª
©
RL = x ∈ X ¯ TX,x ∩ L 6= ∅ .
The Gauss map γ is defined by the linear system |RL | generated by the divisors RL ,
L ∈ G(N, N − n − 1). This linear system does not have fundamental points, and
ramification divisors RL corresponding to various linear subspaces LN −n−1 ⊂ PN
are preimages of Schubert divisors on G(N, n) (cf. [28, Chapter 1; 37, Chapter XIV,
§ 8]).
2.12. Proposition. The linear system |RL | is ample.
Proof. Proposition 2.12 immediately follows from Corollary 2.8 in view of [30,
Chapter II, 6.6.3].
2.13. Remark. In the case when char K = 0 Ein [18] proved that ramification
divisor is ample for an arbitrary nonsingular finite covering of Pn of degree greater
than one.
Let X n ⊂ PN , X n 6= Pn be a nonsingular variety. The exact sequences
0 → TX
N +1
→ OX
→ N (−1) → 0,
0 → OX (−1) → TX
→ ΘX (−1) → 0,
where ΘX is the tangent bundle to X and TX = γ ∗ (S) is the preimage of the
standard vector subbundle S of rank n + 1 on G(N, n) (so that projectivizations of
fibres of TX naturally correspond to projective tangent spaces to X), show that
¡
¢
γ ∗ OG(N,n) (1) ' det TX∗ ' KX (n + 1) = KX ⊗ OX (n + 1),
where KX is the canonical line bundle on X (cf. [64, 6.19]; we denote by the same
symbol a bundle and the corresponding sheaf of sections). We remark that the
property that a section of the line bundle KX (n + 1) vanishes along a divisor from
|RL | lies in the basis of the classical definition of canonical class. An immediate
consequence of Proposition 2.12 is the following
2.14. Corollary. Let X n ⊂ PN , X n 6= Pn be a nonsingular variety. Then
KX (n + 1) is an ample line bundle.
2.15. Remark. It is worthwhile to compare Corollary 2.14 with some known
results on the index of Fano varieties [51]. In general the role of very ampleness
versus ampleness in such type of results is still to be investigated. However in the
conditions of Corollary 2.14 the bundle KX (n + 1) is actually very ample, at least if
26
I. THEOREM ON TANGENCIES AND GAUSS MAPS
char K = 0 (cf. [18]). This is easily shown by induction on n using the fact that X
has sufficiently many nonsingular hyperplane sections, and by Kodaira’s vanishing
theorem, for such a section H n−1 ⊂ X n the complete linear system
|KH + nH 2 | = |KX + (n + 1)H · H|
is cut by the linear system |KX + (n + 1)H| (here KH is the canonical class of H;
we denote by the same symbol the canonical divisor class and the canonical line
bundle).
2.16. Proposition. Let X n ⊂ PN be a nondegenerate variety, and let Y r ⊂
X be a subvariety of X for which m − n = codimL Y <©codimP£N X =
¤ª N − n,
where Lm = h Y i is the linear span of Y . Then r ≤ min n − 1, N2+b , where
b = dim (Sing X).
n
Proof. Without loss of generality we may assume that Y 6⊂ Sing X. From our
assumption it follows that for an arbitrary point y ∈ Y
dim (TX,y ∩ L) ≥ dim TY,y ≥ r.
(2.16.1)
Hence
¯
©
ª
γ(Y ) = γ(Y ∩ Sm X) ⊂ α ∈ G(N, n) ¯ dim Lα ∩ L ≥ r = S(L, r) ⊂ G(N, n),
where S(L, r) is the corresponding Schubert cell and γ : X 99K G(N, n) is the Gauss
map. Since by our assumption m − r < N − n, i.e. n + m − r < N , from (2.16.1) it
follows that for each point y ∈ Y ∩ Sm X there exists a hyperplane M containing
L which is tangent to X at y. Put
¯
©
ª
S(M, L, r) = α ∈ G(N, n) ¯ Lα ∈ M, dim Lα ∩ L ≥ r .
Then S(M, L, r) ⊂ S(L, r) and
dim S(M, L, r) = (r + 1)(m − r) + (n − r)(N − n − 1),
dim S(L, r) = (r + 1)(m − r) + (n − r)N − n),
codimS(L,r) S(M, L, r) = n − r = codimX Y.
©
ª
Replacing if necessary r by min dim (TX,y ∩ L) we may assume that
y∈Y
γ(Y ) ∩ S(M, L, r) ∩ Sm (S(L, r)) 6= ∅.
Then
dim (γ(Y ) ∩ S(M, L, r)) ≥ dim γ(Y ) − codimS(L,r) S(M, L, r)
= (r − f ) − (n − r) = 2r − n − f,
¯
where f is the dimension of general fiber of γ¯Y . On the other hand
γ(Y ) ∩ S(M, L, r) = γ
¯
¡©
ª¢
y ∈ Y ∩ Sm X ¯ TX,y ⊂ M ,
(2.16.2)
(2.16.3)
2. GAUSS MAPS OF PROJECTIVE VARIETIES
27
and from Theorem 1.7 it follows that
dim (γ(Y ) ∩ S(M, L, r)) ≤ N − n + b − f.
(2.16.4)
Combining
£
¤ (2.16.3) and (2.16.4), we conclude that 2r − n − f ≤ N − n + b − f, i.e.
r ≤ N2+b . Proposition 2.16 is proved. ¤
£
¤
We observe that N2+b < n − 1 for N < 2n − b − 2.
2.17. Remark. For K = C, b = −1 Proposition 2.16 can be also deduced from
the Barth-Larsen theorem on the structure of integral cohomology of X (cf. [54]).
2.18. Remark. It is worthwhile to compare Proposition 2.16 with the known
classical result the first rigorous proof of which was probably given by Lluis (cf. [58,
Lema 1, Corolario 1]) in which r is arbitrary, but L is a general linear subspace.
2.19. Example. Let X0n−b−1 , n ≥ b + 5, n + b ≡ 1 (mod 2) be a general linear
projection of the Grassmann variety G( n−b+1
, 1) in P2n−2b−5 , and let X n ⊂ PN ,
2
b
N = 2n − b − 4 be a cone with vertex P and base X0 . Then X0 ' G( n−b+1
, 1)
2
(cf. [33; 38]) and dim (Sing X) = b. Furthermore, X n ⊃ Y r , where Y r , b < r < n,
r ≡ n (mod 2) is the cone with vertex Pb over Y0r−b−1 , and Y0 is the projection of a
Grassmann subvariety G( r−b+1
, 1) ⊂ G( n−b+1
, 1). Then m = b+1+2(r−b−1)−3 =
2
2
2r − b − 4, and m − r = r − b − 4 < N − n = n − b − 4. On
£ the
¤ other hand, for
r = n − 2 we have an equality in Proposition 2.16, viz. r = N2+b = 2n−4
2 .
2.20. Corollary. If£ X n ¤6= Pn , then X does not contain linear subspaces of
dimension greater than N2+b . If X is not a hypersurface (i.e. N > n + 1), then X
£
¤
does not contain projective hypersurfaces of dimension greater than N2+b .
The following examples show that the bound in Corollary 2.20 is sharp.
2.21. Example. a1 ) Let X0n−b−1 , n ≥ b + 2 be a nonsingular quadric, and let
X ⊂ PN , N = n+1 be a cone with vertex Pb and base X0 . Then £dim (Sing
b,¤
¤ X)
£ N=
+b
=
and X contains a linear subspace Y r = Pr , where r = b + 1 + n−b−1
2
2
(cf. [28, Volume 2, Chapter 6; 37]).
a2 ) Let X0 = P1 × Pn−b−2 , b ≥ b + 3 be a Segre variety, and let X n ⊂ PN ,
N = 2n − b − 2 be a cone with vertex Pb and base X0 . Then dim (Sing X) = b, and
X contains a linear subspace Y n−1 = Pn−1 . In this case r = n − 1 = N2+b .
b) In the assumptions of Example 2.19, let n = b + 7. Then X n ⊂ Pn+3
contains the quadratic cone Y n−2 with vertex Pb whose base is a nonsingular fourdimensional quadric G(3, 1). Here n − 2 = n+b+3
= N2+b .
2
n
Apparently, it is hard to construct£ examples
of multi-dimensional varieties con¤
taining a hypersurface of dimension N 2−1 .
28
I. THEOREM ON TANGENCIES AND GAUSS MAPS
3. Subvarieties of complex tori
Besides subvarieties of projective space there is another important class of varieties for which it is natural to introduce Gauss maps, viz. subvarieties of complex
tori.
Let AN be an n-dimensional complex torus, and let X n ⊂ AN be an analytic
subset. Let CN be the universal covering of AN , and let CN → AN be the corresponding homomorphism of abelian groups. Using shifts, one can identify the
tangent space to AN at an arbitrary point z ∈ AN with CN , and the tangent space
to X at a point x ∈ X can be identified with a vector subspace ΘX,x ⊂ CN .
3.1. Definition. Let A be a complex torus, and let Y ⊂ A be a connected
analytic subset. The smallest subtorus of A containing all the differences y − y 0 ,
y, y 0 ∈ Y (in the sense of group structure on A) is called the toroidal hull of Y and
is denoted by h Y i.
We observe that for an arbitrary point y ∈ Y we have Y ⊂ y + h Y i.
3.2. Lemma. Let Y ⊂ AN be a connected compact analytic subset whose
tangent subspaces at smooth points are contained in a vector subspace Cm ⊂ CN .
Then dim h Y i ≤ m.
Proof. It is easy to see that there exist an N -dimensional torus ÃN and an mdimensional subtorus T m ⊂ ÃN , T m ⊃ Y such that à is locally isomorphic to
A in a neighborhood of Y . It is clear that in a suitable neighborhood of T in
à and therefore in sufficiently small neighborhoods of Y in à and Y in A there
exist N − m analytically independent holomorphic functions. On the other hand,
from [5] and [36] it follows that in a small neighborhood of Y in A there exist
exactly dim A − dim h Y i analytically independent holomorphic functions. Hence
dim A − dim h Y i ≥ N − m, i.e. dim h Y i ≤ m. ¤
3.3. Definition. Let X n ⊂ AN be an n-dimensional analytic subset of an N dimensional torus A, and let Y r be an r-dimensional analytic subset of X. We say
that a vector subspace Cm ⊂ CN is tangent to X along an analytic subset Y ⊂ X
if Cm ⊃ ΘX,y for all y ∈ Y .
3.4. Lemma. Let X n ⊂ AN be an analytic subset, and let Cm ⊂ CN be a
vector subspace which is tangent to X along a connected compact analytic subset
Y r ⊂ X n . Then there exist an N -dimensional complex torus ÃN , an m-dimensional
complex subtorus T m ⊂ ÃN , T m ⊃ Y r , neighborhoods U ⊂ AN , U ⊃ Y , U +
h Y iA = U , Ũ ⊂ ÃN , Ũ ⊃ Y , Ũ + h Y ià = Ũ , and an analytic subset X̃ ⊂ T ∩ Ũ
such that Ũ ' U , h Y iA ' h Y ià = h Y iT , and X̃ ' X ∩ U , and the mappings
∼
U → Ũ , T ,→ Ã and X ∩ U ,→ T ∩ Ũ are compatible with the action of h Y i.
Proof. The tori à and T are constructed as in Lemma 3.2. To construct X̃ it
suffices to take the preimage of X in CN and to project it to the universal cover
Cm of the torus T m . Considering the quotient tori, it is easy to verify that this can
be done equivariantly. ¤
3. SUBVARIETIES OF COMPLEX TORI
29
3.5. Theorem. Let X n ⊂ An be an analytic subset of a complex torus, and let
C ⊂ CN be a vector subspace which is tangent to X along a connected compact
analytic subset Y r ⊂ X n . Then for some neighborhood Y ⊂ U ⊂ A we have
X ∩ U ⊂ X̄U , where X̄U is a product of the torus h Y i ⊂ A, dim h Y i = k and a
(local) analytic subset of an (m − k)-dimensional complex torus B m−k , and there
is a natural isomorphism Cm ' Ck × Cm−k , where Ck ⊂ CN is the universal cover
of the torus h Y i and Cm−k is the universal cover of the torus B.
m
Proof. In the notations of Lemma 3.4 we consider the canonical holomorphic
mappings
π : A → A/h Y iA ,
π̃ : à → Ã/h Y ià ,
πT : T → T /h Y iT .
From Lemma 3.4 it follows that
XU0 = π(X ∩ U ) ' π̃(X̃) ' πT (X̃),
where
π(Y ) = y ∈ XU0 ⊂ X 0 = π(X).
In particular, the neighborhood XU0 of the point y in X 0 embeds as an analytic
subset in the (m − k)-dimensional torus B = T /hY iT . We put
X̄ = π −1 (X 0 ) ⊂ A,
X̄U = X̄ ∩ U = π −1 (XU0 ).
Then X̄U is the desired analytic subset of U , and for an arbitrary point z ∈ X̄U
the tangent space to X̄U at z has dimension not exceeding m and is tangent to X
along the analytic subset
X ∩ π −1 (π(z)) = X ∩ (z + h Y iA ) .
¤
3.6. Corollary (Theorem on tangencies for subvarieties of complex
tori). Let X n ⊂ AN be an analytic subset of a complex torus, and let Cm ⊂ CN be
a vector subspace which is tangent to X along a compact analytic subset Y r ⊂ X n .
Then r ≤ km , where km is the maximal dimension of complex subtorus C ⊂ A such
that dim (X + C) ≤ m.
3.7. Remark. In contrast to the case of subvarieties of projective spaces (cf.
Corollary 1.8), in Corollary 3.6 we do not assume that X is nondegenerate (an
analytic subset X ⊂ A is called nondegenerate if hXi = A). However if dim X ≤ m,
then k ≥ dim hXi ≥ n and Corollary 3.6 is trivial.
Let X n ⊂ AN be an analytic subset of a complex torus, let n ≤ m ≤ N − 1, and
let
¯
©
ª
P = (x, α) ∈ Sm X × Gras (N, m) ¯ Lα ⊃ ΘX,x ,
where Gras (N, m) ' G(N − 1, m − 1) is the Grassmann variety of m-dimensional
N
vector subspaces in CN , Lm
is the vector subspace corresponding to a point
α ⊂C
α ∈ Gras (N, m), and the bar denotes closure in X × Gras (N, m). We denote by
pm : Pm → X (resp. γm : Pm → Gras (N, m)) the projection map to the first (resp.
second) factor.
30
I. THEOREM ON TANGENCIES AND GAUSS MAPS
3.8. Definition. The mapping γm is called the mth Gauss map, and its image
∗
Xm
= γm (Pm ) ⊂ Gras (N, m) is called the variety of tangent m-spaces to the
variety X.
In particular, for m = n we obtain the usual Gauss map γ : X 99K Gras (N, n),
N −1
N −1
and for m = N − 1 we get a map γN −1 : PN
.
−1 → P
3.9. Proposition. Let X n ⊂ AN be an irreducible compact analytic subset.
Then there exists an analytic subtorus C k ⊂ AN such that
(i) X + C = ¯C;
(ii) γ = γ 0 ◦π¯X , where π : A → B, B = A/C is the canonical holomorphic
map and γ : X 99K Gras (N, n) and γ 0 : X 0 99K Gras (N − k, n − k), X 0 =
π(X) ⊂ B are the Gauss maps;
(iii) the map γ 0 : X 0 99K γ 0 (X 0 ) ⊂ Gras (N − k, n − k) is generically finite.
Proof. Arguing by induction, we assume that Proposition 3.9 is already verified
for N 0 < N and prove it in the case dim A = N . If the map γ is generically
finite, then it suffices to put C = 0, X 0 = X. Suppose that for a general point
x ∈ X we have dim γ −1 (γ(x)) > 0, and let Y be a positive-dimensional component
of γ −1 (γ(x)). By Lemma 3.2 0 < k = dim h Y i ≤ n. Since a continuous family of
complex analytic subtori of A is constant, we conclude that if x̃ is another general
point of X and Ỹ is a positive-dimensional component of the fiber γ −1 (γ(x̃)), then
h Ỹ i = h Y i. We put
C = h Y i,
X 0 = π(X) ⊂ B,
B = A/C,
x0 = π(Y ) = π(x).
Since the tangent
¡ ¯ ¢space to X is constant along Y ∩ Sm X and the kernel of the
differential dx π¯X coincides with Θπ−1 (x0 ),x , we see that Y lies in a fiber of the
Gauss map for the subvariety π −1 (x0 ) ⊂ C. But Y spans C and dim C ≤ n < N
(otherwise Y = X = C = A and Proposition 3.9 is obvious), so that from the
induction hypothesis it follows that Y = C.
¯
Thus a general and therefore each fiber of the map π¯X coincides with the corresponding fiber of the map π : A → B; moreover, X + C = C and X is a locally
trivial analytic fiber bundle over X 0 with fiber C. Furthermore,
Sing X = π −1 (Sing X 0 ),
ΘX,x = ΘX 0 ,x0 ⊕ Ck ,
¯
where Ck ⊂ CN is the universal covering of C and γ = γ 0 ◦π¯X .
¤
3.10. Corollary. Let X n ⊂ AN be a compact complex submanifold. Then
the Gauss map γ : X → Gras (N, n) can be represented in the form γ = γ 0 ◦π,
where π : X → X 0 is a locally trivial analytic fiber bundle whose fiber is a complex
subtorus C k ⊂ AN , X 0 is a compact complex subvariety of the torus B = A/C,
and the Gauss map γ 0 : X 0 → Gras (N − k, n − k) is finite. In particular, if X does
3. SUBVARIETIES OF COMPLEX TORI
31
not contain complex subtori (e.g. if A is a simple torus), then the Gauss map γ is
finite.
Proof. Corollary 3.10 is an immediate consequence of Theorem 3.5 and Proposition 3.9. ¤
Our results also allow to describe the structure of Gauss maps γm for arbitrary
n ≤ m ≤ N − 1.
3.11. Theorem. Let X n ⊂ AN be a compact analytic submanifold, n ≤ m ≤
N − 1. Then
a) there exist finitely many subtori C1 , . . . , Cl ⊂ A such that if X̄i = X + Ci ,
∗
N
i = 1, . . . , l, α ∈ Xm
, Lα is the m-dimensional vector subspace
¡ −1 ¢of C
corresponding to α, and Y is a connected component of pm γm (α) , then
for some 1 ≤ i¡ ≤ l we have h Y i = Ci , L¡α is¢ tangent¡ to X̄i along
¢¢ a torus
∗
y + Ci , y ∈ Y so that in particular α ∈ X̄i m = γm Pm (X̄i ) , and Y is
a connected component of the analytic subset X ∩ (y + Ci );
b) the components of general fibers of the Gauss maps are ¡the same.
¢ More
precisely, if n ≤ m, m0 ≤ N − 1 and x ∈ X, αm ∈ γm p−1
(x)
, αm0 ∈
m
¡
¢
γm0 p−1
(x)
are
general
points,
then
in
a
neighborhood
of
x
we
have
m0
¡ −1
¢
¡ −1
¢
pm γm
(αm ) = pm0 γm
0 (αm0 ) .
If C ⊂ X is the maximal analytic subtorus of A for which X + C = X,
∗
then a general subspace Lα ⊂ CN , α ∈ Xm
is tangent to X along a union
of tori of the form x + C, x ∈ X.
Proof. Theorem 3.11 is an immediate consequence of Theorem 3.5 and Corollary 3.10. ¤
3.12. Corollary. If X does not contain complex subtori, then for an arbitrary
n ≤ m ≤ N − 1 the mth Gauss map γm is generically finite. In particular,
¡
¢
∗
dim Xm
= dim Pm = n + dim Gras (N − n, m − n) = n + (m − n)(N − m)
∗
N −1
(compare with (2.3.1)) and XN
. If A is a simple torus (i.e. A does not
−1 = P
∗
contain proper analytic subtori), then all Gauss maps γm : Pm → Xm
are finite.
3.13. Let X n ⊂ AN be an analytic submanifold. Then the tangent bundle
ΘX naturally embeds in the restriction of the tangent bundle ΘA on X (which
N
is a trivial¡ bundle
¯ ¢± on X with fiber C ), and we can consider the normal bundle
NA/X = ΘA¯X ΘX . It is clear that if S (resp. Q) is the canonical vector sub(resp. quotient-) bundle on Gras (N, n) and γ : X → Gras (N, n) is the Gauss map,
then ΘX = γ ∗ (S) and NA/X = γ ∗ (Q). In other words, the Gauss map γ is induced
by the normal bundle NA/X and the linear map
Γ(A, ΘA ) → Γ(X, NA/X )
of the corresponding vector spaces
of
(cf. [28, Volume 1, Chapter I, § 5]).
¢ sections
¡
N −1
→
P
is induced by the invertible sheaf
Similarly, the
map
γ
:
P
N
N
−1
A/X
¡
¢
ON (1) on P NA/X = PN −1 .
32
I. THEOREM ON TANGENCIES AND GAUSS MAPS
The exact sequence
¯
0 → ΘX → ΘA¯X → NA/X → 0
shows that
det NA/X = − det ΘX = KX ,
where KX is the canonical line bundle on X. Since for the Plücker embedding we
have det Q = OGras (N,n) (1), the map γ is also defined by a (base point free) linear
subsystem of the canonical linear system |KX |, viz. by the linear system spanned
by the ramification divisors
¯
©
ª
RL = x ∈ X ¯ dim (ΘX,x ∩ L) > 0 ,
where L runs through the set of general (N − n)-dimensional vector subspaces of
CN (compare with Section 2).
3.14. Proposition. Let X n ⊂ AN be an analytic submanifold.
a) The following conditions are equivalent:
(i) The bundle NA/X is ample;
(ii) The mappings γm , n ≤ m ≤ ¡N − 1 ¢are finite;
∗
N −1
(iii) XN
and γN −1 : P NA/X → PN −1 is a finite covering.
−1 = P
b) Suppose that condition
(iii) from a) holds. Then either n = N − 1 or
¡ ¢
deg γN −1 = cn Ω1X = (−1)n cn (X) = |e(x)| ≥ N − 1, where e(X) is the
(topological) Euler-Poincaré characteristic of X and Ω1X is the sheaf of
differential forms of rank one.
Proof. a) (i)⇔(iii) in view of the definition of ampleness of vector bundle (cf.
[31, Chapter III]), Corollary 6.6.3 from [31, Chapter II] and Proposition 2.6.2 from
[30, Chapter III1 ], (ii)⇒(iii) is obvious, and (iii)⇒(ii) follows from the fact that
for m < N − 1 the fibers of γm (or, more precisely, their projections to X) are
contained in the fibers of γN −1 .
b) From the description of the map γN −1 given in 3.13 it immediately follows
that
¡ ¢
deg γN −1 = cn (Θ∗X ) = cn Ω1X = (−1)n cn (X) = |e(X)|.
In [55, 3.1] it is shown that if Y is a complex manifold and π : Y → Pk is a
finite covering of degree¡ ≤ k −
¢ 1, then Pic Y = Z. To verify b) it suffices to
put¡ k =
N
−
1,
Y
=
P
N
and to observe that for N − n − 1 > 0 we have
A/X
¡ ¡
¢¢¢
rk Pic P NA/X
≥ 2.
¤
We observe that in view of Corollary 3.12 assertions (i)–(iii) hold in the case
when A is a simple torus.
3.15. Proposition. Let X n ⊂ AN be an analytic submanifold. Then the
canonical linear system |KX | is base point free, and its suitable multiple defines a
holomorphic mapping π : X → X 0 making X a locally trivial analytic fiber bundle
over a complex manifold X 0 ; the fiber of π is the maximal analytic subtorus C ⊂ A
for which X + C = X (where X 0 embeds isomorphically in B = A/C).
Proof. In view of the above description of Gauss map (cf. 3.13), Proposition 3.15
immediately follows from Corollary 3.10 and Corollary 6.6.3 from [30, Chapter II].
3. SUBVARIETIES OF COMPLEX TORI
33
3.16. Corollary. Let X n ⊂ AN be a nondegenerate complex submanifold (i.e.
hXi = A). Then there exists an analytic subtorus C ⊂ A such that if π : A → B =
A/C is the projection map, then
¯
(i) π¯ : X → X 0 ⊂ B is a locally trivial analytic fiber bundle with fiber C (so
X
0
that X = π −1 (X
¯ ));
¯
(ii) the mapping π X is equivalent to the mapping defined by a sufficiently high
multiple of the canonical class KX ;
(iii) the canonical class KX 0 is ample;
(iv) B = hX 0 i is an abelian variety.
3.17. Corollary. An analytic submanifold X n ⊂ AN is a variety of general
type (i.e. the canonical dimension of X coincides with its dimension) if and only if
the canonical class KX is ample.
3.18. Remark. From Corollary 2.8 it follows that for a nonsingular variety
X n 6= Pn over an algebraically closed field of characteristic zero the Gauss map γ is
birational, and according to Remark 2.15, the map defined by the complete linear
system |KX + (n + 1)H|, where H is a hyperplane section of X, is an isomorphism.
However for submanifolds of complex tori the map γ and the canonical map defined
by the complete linear system of canonical divisors can be finite maps of degree
greater than one. As an example, it suffices to consider a hyperelliptic curve X
of genus g > 1 embedded in its Jacobian variety JX . In this case the Gauss map
coincides with the canonical map which clearly has degree two (it is clear that the
normal bundle NJX /X is ample, and all the Gauss maps γm , 1 ≤ m ≤ g−1 are finite;
cf. Proposition 3.14). In [83] it is shown that in the conditions of Proposition 3.14
deg γX ≤ |e(X)|
N −n .
3.19. Remark. The study of submanifolds of complex tori was begun by Hartshorne [32] and continued by Sommese [84] who revealed the role of complex subtori
using the notion of k-ampleness. At the same time Ueno [93, § 10] undertook a
thorough investigation of properties of the canonical dimension of submanifolds of
complex tori (his results easily follow from ours, but are stated in different terms)
and announced in [92] our Corollary 3.17, but his proof turned out to be erroneous (cf. [93, 10.13]). Griffiths and Harris [29, §4 b)] showed that the map γ 0 from
our Corollary 3.10 is generically finite, and basing on their result Ran [68] gave a
different proof of Corollary 3.17 and of Proposition 3.20 below in the case c = 0.
The following two results are analogs of Proposition 2.16 for submanifolds of
complex tori.
r
3.20. Proposition. Let X n ⊂ AN be
h a complex
i submanifold, and let Y ⊂
−n)+c
X n be a complex subtorus. Then r ≤ n(N
, where c is the maximum of
N −n+1
dimensions of complex subtori C ⊂ A such that X +C = X (this bound is nontrivial
for c < 2n−N −1). In particular, if X is a hypersurface (i.e. n = N −1) containing a
complex subtorus Y r of dimension r > n2 , then X is a locally trivial analytic bundle
whose fiber is a complex torus and whose base is a hypersurface in a complex torus
of smaller dimension.
Proof. It is clear that for an arbitrary point y ∈ Y we have ΘX,y ⊃ ΘY,y = Cr ,
34
I. THEOREM ON TANGENCIES AND GAUSS MAPS
where Cr ⊂ CN is the universal covering of the torus Y . Hence
¯
©
ª
γX (Y ) ⊂ SY = α ∈ Gras (N, n) ¯ Lα ⊃ Cr
and by Corollary 3.10
r − c ≤ dim γX (Y ) ≤ dim SY = dim (Gras (N − r, n − r)) = (n − r)(N − n)
which implies the assertion of the proposition. ¤
3.21. Proposition. Let X n ⊂ AN be a complex submanifold, and let Y r ⊂ X n
be an analytic subset for which dim h Y i = m, where m − r = codim h Y i Y <
codimA X = N − n. Denote by d the maximal
£ n+d ¤ dimension of complex subtori
D ⊂ A for which
X
+
D
=
6
A.
Then
r
≤
. In particular, if A is a simple
2
£n¤
torus, then r ≤ 2 .
Proof. Proposition 3.21 can be proved in essentially the same way as Proposition 2.16. In the notations corresponding to those of 2.16 we have
¡
¢
dim γ(Y ) ∩ S(M, L, r) ≥ dim γ(Y ) − codimS(L,r) S(M, L, r)
= (r − f ) − (n − r) = 2r − n − f,
(3.21.1)
¯
where f is the dimension of general fiber of γ¯Y (compare with (2.16.2)). On the
other hand, from Corollary 3.6 it follows that
dim (γ(Y ) ∩ S(M, L, r)) ≤ d − f.
Combining (3.21.1) and (3.21.2) we get 2r − n − f ≤ d − f, i.e. r ≤
required. ¤
£
¤
We observe that n+d
< n − 1 for d < n − 2.
2
(3.21.2)
£ n+d ¤
2
as
3.22. Corollary. If£ X ¤6= A, then X does not contain complex subtori of
dimension greater than n+d
. If X is not a hypersurface (i.e. N > n + 1), then X
2
¤
£
.
does not contain hypersurfaces (in complex tori) of dimension greater than n+d
2
3.23. Remark. In contrast to the case of subvarieties of projective spaces (cf.
Proposition 2.16), in Proposition 3.21 we do not assume that X is nondegenerate.
However if hXi 6= A, then d ≥ dim hXi ≥ n, so that in the degenerate case our
results are trivial.
CHAPTER II
PROJECTIONS OF ALGEBRAIC VARIETIES
Typeset by AMS-TEX
35
36
II. PROJECTIONS OF ALGEBRAIC VARIETIES
1. An existence criterion for good projections
Let Y r ⊂ X n be a nonempty irreducible r-dimensional subvariety of an irreducible n-dimensional variety X defined over an algebraically closed field K, and
let ∆Y ⊂ Y × Y ⊂ Y × X be the diagonal. Denote by IY the Ideal of ∆Y in Y × X
and put
³∞ ±
´
0
ΘY,X
= Spec ⊕ I j I j+1 ,
j=0
0
Θ0Y,X,y = ΘY,X
⊗ K(y),
y ∈ Y.
1.1. Definition. We call Θ0Y,X,y the (affine) tangent star to X with respect to
Y ⊂ X at the point y ∈ Y .
It is easy to see that
Θ0y,X ⊂ Θ0Y,X,y ⊂ Θ0X,y ⊂ ΘX,y ,
where Θ0y,X = Θ0y,X,y is the (affine) tangent cone to X at the point y, Θ0X,y = Θ0X,X,y
is the (affine) tangent star to X at y, and ΘX,y is the Zariski tangent space to X
at y. Furthermore, if X n ⊂ PN and the bar denotes projective closure, then in the
notations of Section 1 of Chapter 1 we have
0
Θ0y,X = Ty,X
,
0
Θ0Y,X,y = TY,X,y
,
0
,
Θ0X,y = TX,y
ΘX,y = TX,y
(cf. [45]).
1.2. Definition. Let f : X → X 0 be a morphism of algebraic varieties. We say
that f is unramified in the sense of Johnson
(J-unramified) with respect to Y ⊂ X
¯
at a point y ∈ Y if the morphism dy f ¯Θ0
is quasifinite. If f is J-unramified with
Y,X,y
respect to Y at all points y ∈ Y , then we say that f is J-unramified with respect
to Y . If moreover Y = X, then the morphism f is called J-unramified.
1.3. Definition. In the notations of Definition 1.2 we say that f is an embedding in the sense of Johnson (J-embedding) with respect to Y ⊂ X if f is Junramified with respect to Y and is one-to-one on f −1 (f (Y )). If moreover Y = X,
then the morphism f is called J-embedding.
1.4. Remark. If X is nonsingular along Y, i.e. Y ∩ Sing X = ∅ and Y ⊂ Sm X,
then f is unramified with respect to Y if and only if f is unramified at all points
y ∈ Y ; f is a J-embedding with respect to Y if and only if f is a closed embedding
in some neighborhood of Y in X.
1.5. Proposition. Let X n ⊂ PN be a projective algebraic variety, let Y r ⊂ X n
be a nonempty irreducible subvariety, let LN −m−1 ⊂ PN , L ∩ X = ∅ be a linear
subspace, and let π : X → Pm be the projection with center in L.
a) The following conditions are equivalent:
(i) The morphism π is J-unramified with respect to Y ;
(ii) L ∩ T 0 Y, X) = ∅.
1. AN EXISTENCE CRITERION FOR GOOD PROJECTIONS
37
b) The following conditions are equivalent:
(i) The morphism π is unramified at the points of Y ;
(ii) L ∩ T (Y, X) = ∅.
c) The following conditions are equivalent:
(i) The morphism π is a J-embedding with respect to Y ;
(ii) L ∩ S(Y, X) = ∅.
d) The following conditions are equivalent:
(i) The morphism π is an isomorphic embedding;
(ii) L ∩ S(Y, X) = L ∩ T (Y, X) = ∅.
Proof. Most of the assertions
of the proposition ¯are obvious. To verify a) it
¯
is finite or equivalently
is quasifinite iff π¯T 0
suffices to use the fact that π¯Θ0
Y,X,y
Y,X,y
0
0
is a projective cone with vertex y). ¤
= ∅ (we recall that TY,X,y
L ∩ TY,X,y
1.6. Proposition. a) In the conditions of Proposition 1.5 suppose that the
morphism π : X n → Pm is J-unramified with respect to an irreducible
subvariety Y r ⊂ X n , where m < r + n (i.e. dim L ≥ N − n − r). Then π
is a J-embedding with respect to Y .
b) In the conditions of Proposition 1.5 suppose that the morphism π : X n →
Pm is unramified at all points y ∈ Y r , where Y r ⊂ X n is an irreducible
subvariety and m < r+n (i.e. dim L ≥ N −n−r). Then π is an isomorphism
in a neighborhood of Y .
Proof. In view of Proposition 1.5 a), our condition means that
Therefore
L ∩ T 0 (Y, X) = ∅.
(1.6.1)
dim T 0 (Y, X) < codimPN L ≤ r + n.
(1.6.2)
In view of Theorem 1.4 of Chapter I, from (1.6.2) it follows that
S(Y, X) = T 0 (Y, X).
(1.6.3)
In view of Proposition 1.5 c), assertion a) of Proposition 1.6 now follows from (1.6.1)
and (1.6.3).
b) According to Proposition 1.5 b), our condition means that
Therefore
L ∩ T (Y, X) = ∅.
(1.6.4)
dim T 0 (Y, X) ≤ dim T (Y, X) < codimPN L ≤ r + n.
(1.6.5)
By Theorem 1.4 of Chapter I, from (1.6.5) it follows that
S(Y, X) = T 0 (Y, X).
(1.6.6)
In view of Proposition 1.5 d), our assertion now follows from (1.6.4), (1.6.6), and
the obvious inclusion T 0 (Y, X) ⊂ T (Y, X).
¤
38
II. PROJECTIONS OF ALGEBRAIC VARIETIES
1.7. Corollary. Let X n ⊂ PN be a projective variety, let LN −m−1 ⊂ PN ,
L ∩ X = ∅ be a linear subspace, and let π : X → Pm be the projection with center
at L. Suppose that m ≤ 2n − 1. Then
a) The morphism π is J-unramified if and only if π is a J-embedding;
b) The morphism π is unramified if and only if π is an embedding.
1.8. Remark. Corollary 1.7 was proved by Johnson [45] by means of formal
computations involving characteristic classes under the assumption N ≤ 2n. For
nonsingular varieties Corollary 1.7 was proved in [26], and the general case was
settled in [27, § 5] and [97, § 2] (in these papers the authors actually consider various
special cases of Theorem 1.4 of Chapter I for Y = X).
1.9. Lemma. Let X ⊂ PN be a nondegenerate variety, x ∈ X, y ∈ PN , y 6= x,
z ∈ hy, xi, z 6= y. Then
a) TS(y,X),y = PN ;
b) TS(y,X),z ⊃ hy, TX,x i;
¡
¢
c) dy×x×z ϕy ΘSy,X ,y×x×z = hy, TX,x i, where S(y, X) is the cone with vertex
y and base X (cf. Section 1 of Chapter I), dϕy is the differential of the map
ϕy , and the bar denotes closure in the Zariski topology.
Proof. a) It is clear that TS(y,X),y ⊃ S(y, X) ⊃ X. Therefore hXi ⊂ TS(y,X),y .
But by definition for a nondegenerate variety X ⊂ PN we have hXi = PN .
b) It suffices to consider the affine case. Furthermore, we may assume that y
coincides with the origin. Since the affine part of S(y, X) is a cone, the (embedded)
tangent spaces at the points z and µz (µ ∈ K ∗ = K \ 0) coincide with each other
and contain the origin. To verify b) it suffices to choose µ so that µz = x.
c) As in the proof of b), we consider the affine case and assume that y coincides
with the origin. Then the restriction of py2 on the affine part of Sy,X admits a
section σ,
σ(x0 ) = (y, x0 , λx0 ),
x0 ∈ X,
z = λx
and
¡
¢
dy×x×z ϕy ΘSy,X ,y×x×z
D
¢
¡
¢E
¡
= dy×x×z ϕy Θ(py )−1 (y×x),y×x×z , dy×x×z ϕy Θσ(X),y×x×z
12
= h0, x, ΘλX,z i = h0, ΘX,x i
which implies c).
¤
1.10. Proposition. a) Let y ∈ Y ⊂ X ⊂ PN , x ∈ X, x 6= y, z ∈ hy, xi. Then
TS(Y,X),z ⊃ hTY,y , TX,x i.
b) Suppose in addition that char K = 0. Then for general points y ∈ Y ,
x ∈ X, z ∈ hy, xi we have TS(Y,X),z = hTY,y , TX,x i .
1. AN EXISTENCE CRITERION FOR GOOD PROJECTIONS
39
Proof. a) It is clear that
S(Y, X) ⊃ S(y, X),
S(Y, X) ⊃ S(x, Y ).
Therefore
­
®
TS(Y,X),z ⊃ TS(y,X),z , TS(x,Y ),z .
(1.10.1)
According to statements a) and b) of Lemma 1.6,
TS(y,X),z ⊃ TX,x ,
TS(x,Y ),z ⊃ TY,y .
(1.10.2)
Assertion a) immediately follows from (1.10.1) and (1.10.2).
b) First of all we observe that if z ∈ hy, xi, then in the notations of Section 1 of
Chapter 1
ΘSY,X ,y×x×z
¿
À
¡
¢
¡
¢
= Θ Y −1
, Θ Y −1
p1
(y),y×x×z
p2
(x),y×x×z
­
®
= ΘSy,X ,y×x×z , ΘSY,x ,y×x×z .
(1.10.3)
From Lemma 1.9 c) it follows that
¡
¢
dy×x×z ϕY ΘSY,X ,y×x×z = hTY,y , TX,x i .
(1.10.4)
If char K = 0, then the map
dy×x×z ϕY : ΘSY,X ,y×x×z → ΘS(Y,X),z
is surjective for a general point y ×x×z ∈ SY,X , and assertion b) follows from (1.10.3)
and (1.10.4).
¤
1.11. Remark. The arguments used in the proof of Lemma 1.9 also work in
a slightly more general situation. In particular, in Chapter IV we shall need the
0
following variant of Lemma 1.9 b): for x ∈ Sm X we have Tz,(y,X)
⊃ TX,x (we
0
recall that, in the notations of Section 1 of Chapter 1, Tz,S(y,X) is the tangent
cone to S(y, X) at the point z. Similarly, if in the conditions of Proposition 1.10
0
x, y ∈ Sm X, then Tz,S(Y,X)
⊃ TY,y , TX,x .
1.12. Remark. Roberts showed (cf. [70]) that for each prime number p > 0 there
exists an irreducible (singular) curve X ⊂ P3K , char K = p such that for Y = X
and a general point z ∈ P3 = SX the inclusion in statement a) of Proposition 1.10
is strict. An example of such curve is given by the projective closure of the image
2
of affine line A1K under the embedding t 7→ (t, tp , tp ).
40
II. PROJECTIONS OF ALGEBRAIC VARIETIES
1.13. Theorem. Let X n ⊂ PN be a projective variety, and let Y r ⊂ X n be an
irreducible subvariety. Consider the following conditions:
a) For an arbitrary linear subspace LN −m−1 ⊂ PN the projection X → Pm
with center at L is a J-embedding with respect to Y ;
b) There exists a linear subspace LN −m−1 ⊂ PN such that the projection
X → Pm with center at L is a J-embedding with respect to Y ;
c) dim S(Y, X) ≤ m;
d) There exists a Zariski open subset U ⊂ Y × X such that for y × x ∈ U we
have dim hTY,y , TX,x i ≤ m;
e) For all points y ∈ Sm Y , x ∈ Sm X we have dim hTY,y , TX,x i ≤ m.
Then a) ⇔ b) ⇔ c) ⇒ d) ⇔ e). If in addition char K = 0, then all conditions a)–e)
are equivalent to each other.
Proof. From Proposition 1.5 c) it immediately follows that a) ⇒ b) ⇒ c) ⇒ a).
c) ⇒ d). In the notations of Section 1 of Chapter I we set
¡
¢
U = pY12 (ϕY )−1 (Sm S(Y, X)) \ ∆Y .
(1.13.1)
It is clear that U is a Zariski open subset in Y ×X. Let y × x ∈ U . In view of
(1.13.1) there exists a point z ∈ hy, xi ∩ Sm S(Y, X). From Proposition 1.10 a) it
follows that TS(Y,X),z ⊃ hTY,y , TX,x i . Hence
dim hTY,y , TX,x i ≤ dim S(Y, X) ≤ m.
d) ⇒ e). It is easy to see that the function
¡
¢
y ×x 7→ dim TY,y ∩ TX,x
is upper semicontinuous on Y ×X. Since the functions y 7→ dim TY,y , x 7→ dim TX,x
are constant on Sm Y and Sm X (and are equal to r and n respectively, the function
¡
¢
s(y, x) = dim hTY,y , TX,x i = dim TY,y + dim TX,x − dim TY,y ∩ TX,x
is lower semicontinuous on Sm Y × Sm X. From d) it follows that s(y, x) ≤ m for
y × x ∈ U . Hence from semicontinuity of s and irreducibility of Sm Y × Sm X it
follows that s(y, x) ≤ m for all y ∈ Sm Y , x ∈ Sm X.
e) ⇒ d) is obvious; it suffices to set
U = Sm Y × Sm X.
Next we verify the implication e) ⇒ c) under the assumption that char K = 0.
It is clear that there exist points y ∈ Sm Y , x ∈ Sm X, z ∈ hy, xi such that y × x × z
is a general point of SY,X in the sense of proposition 1.10 b), i.e. the differential
dy×x×z is a surjective map. From Proposition 1.10 b) it follows that
dim S(Y, X) ≤ dim TS(Y,X),z = dim hTY,y , TX,x i ≤ m.
¤
1.14. Corollary. Let X n ⊂ PN be a nonsingular variety over an algebraically
closed field K of characteristic zero. Then X can be isomorphically projected to a
projective space of smaller dimension if and only if for a general (and hence each)
pair of points of X there exists a hyperplane which is tangent to X at these points.
1.15. Remark. Griffiths and Harris [29] proved Corollary 1.14 in the special case
when N ≥ 2n + 1. However Landman discovered that in this case Corollary 1.14
was already proved by Terracini [90].
HARTSHORNE’S CONJECTURE ON LINEAR NORMALITY
41
2. Hartshorne’s conjecture on linear normality and its relative analogs
2.1. Theorem. Let X n ⊂ PN be a nondegenerate variety, and let Y r ⊂ X n
be a closed subvariety. Suppose that there exists a point u ∈ PN \ X such that
the projection π : X → PN −1 with center at u is a J-embedding with respect to Y .
Then codimPN X n = N − n ≥ 12 (r − b) + 1, where b = dim (Y ∩ Sing X).
Proof. Clearly it suffices to consider the case when Y is irreducible. From Proposition 1.5 c) it follows that S(Y, X) 6= PN . Let s = dim S(Y, X), and let z be a
general point of S(Y, X). In the notations of Section 1 of Chapter I we put
¡
¢
L = TS(Y,X),z ,
Qz = pY1 (ψ Y )−1 (z) .
From Theorem 1.4 of Chapter I it follows that either T 0 (Y, X) = S(Y, X) or s =
n + r + 1. In the last case
codimPN
N ≥ s + 1 = n + r + 2,
1
X n = N − n ≥ r + 2 > (r − b) + 1.
2
Thus we may assume that
T 0 (Y, X) = S(Y, X),
Qz 6= ∅,
dim Qz = r + n − s.
It is clear that
L ⊃ TX,x
Let M
N −b−1
⊂P
N
∀x ∈ Qz \ Sing X.
(2.1.1)
be a general linear subspace of codimension b + 1, and let
X 0 = X ∩ M,
Q0z = Qz ∩ M,
Y 0 = Y ∩ M,
L0 = L ∩ M.
Then the variety X 0 ⊂ PN −b−1 is nonsingular along Y 0 , and from (2.1.1) it follows
that
T 0 (Q0z , X 0 ) = T (Q0z , X 0 ) ⊂ L0 .
However it is clear that X 0 6⊂ L0 .Hence
S(Q0z , X 0 ) 6= T 0 (Q0z , X 0 ).
(2.1.2)
From (2.1.2) and Theorem 1.4 of Chapter I it follows that
dim S(Q0z , X 0 ) = dim Q0z + dim X 0 + 1
= (r + n − s − b − 1) + (n − b − 1) + 1
= 2n + r − s − 2b − 1.
(2.1.3)
On the other hand,
¡
¢
dim S(Q0z , X 0 ) ≤ dim S(Y, X) ∩ M = s − b − 1.
(2.1.4)
From (2.1.3) and (2.1.4) it follows that
2n + r − s − 2b − 1 ≤ s − b − 1,
2s ≥ 2n + r − b,
i.e.
codimPN X n = N − n ≥
¤
2N ≥ 2s + 2 ≥ 2n + r − b + 2,
1
(r − b) + 1.
2
42
II. PROJECTIONS OF ALGEBRAIC VARIETIES
2.2. Corollary. Let Y r ⊂ X n ⊂ PN , where X is nonsingular in a neighborhood
of Y . Suppose that there is a point u ∈ PN \ X such that the projection π : X →
PN −1 with center at u is a closed embedding in a neighborhood of Y . Then N ≥
n + 12 (r + 3).
2.3. Remark. From Proposition 1.6 a) it follows that for N < n + r theorem 2.1
(resp. Corollary 2.2) is true if instead of assuming that π is a J-embedding with
respect to Y (resp. an embedding in a neighborhood of Y ) we assume that π is
J-unramified with respect to Y (resp. unramified at all points y ∈ Y ).
The following examples show that the bounds in Corollary 2.2 and Theorem 2.1
are sharp.
2.4. Example. To simplify arguments, in the following examples we assume
that char K = 0.
a) Let X 2 ⊂ P4 be the rational surface F1 of degree three. Let Y 1 ⊂ X 2 be the
minimal section, so that Y is an exceptional curve of the first kind. Then Y ⊂ P4
is a projective line, and the embedding X ,→ P4 is defined by the complete linear
system |Y + 2F |, where F 1 ⊂ X 2 is a fiber of the ruled surface F1 . Since F ⊂ P4
is a projective line, the tangent plane at an arbitrary point of X contains the fiber
passing through this point and therefore intersects Y . From Proposition 1.10 b) it
follows that
dim S(Y, X) = r + n = n + 21 (r + 1) = 3.
Hence
S(Y, X) = T (Y, X) 6= P4
and by Proposition 1.5 d) there exists a projection π : X → P3 which is an isomorphic embedding in a neighborhood of Y (in a suitable coordinate system π(X) ⊂ P3
is defined by the equation u0 u23 = u1 u22 ). In this example
N = 4 = n + 12 (r + 3),
i.e. the inequality in Corollary 2.2 turns into equality.
b) Let X 6 = G(4, 1) ⊂ P9 (the Plücker embedding), and let Y = P3 be the
linear subspace of lines passing through a fixed point of P4 . Then for general points
y ∈ Y , x ∈ X the line TY,y ∩ TX,x parametrizes lines in P4 passing through a fixed
point and intersecting a fixed line. From Proposition 1.10 b) it follows that
dim S(Y, X) = dim T (Y, X) = 8 = r + n − 1 = n + 12 (r + 1) = N − 1,
so that again the inequality in Corollary 2.2 turns into equality.
To show that the bound given in Theorem 2.1 is also sharp for b > −1 it suffices to consider the cone with vertex Pb over one of the varieties constructed in
Example 2.4.
Let X n ⊂ PN be a nonsingular variety, and let Dm (resp. Rm ) be the double
point (resp. ramification) locus with respect to a general projection PN 99K Pm ,
n ≤ m ≤ N − 1. In other words, if LN −m−1 ⊂ PN is a general linear subspace, then
¯
©
ª
Dm = x ∈ X ¯ hx, x0 i ∩ L 6= ∅ ∃ x0 ∈ X \ x ,
¯
©
ª
Rm = x ∈ X ¯ TX,x ∩ L 6= ∅ .
HARTSHORNE’S CONJECTURE ON LINEAR NORMALITY
43
2.5. Corollary. Let r ≥ 2(m − n). Then Y r ∩ Dm 6= ∅ for an arbitrary
subvariety Y r ⊂ X n . If in addition r > 0, then Y r ∩ Rm 6= ∅ for an arbitrary
subvariety Y r ⊂ X n .
Proof. Suppose that Y r ∩ Dm = ∅ (resp. Y r ∩ Rm = ∅). Then for a general
linear subspace LN −m−1 ⊂ PN we have S(Y, X) ∩ L = ∅ (resp. T (Y, X) ∩ L = ∅),
and therefore dim S(Y, X) ≤ m (resp. dim T (Y, X) ≤ m) (compare with Proposition 1.5 b), d)). On the other hand, from Theorem 2.1 it follows that
s = dim S(Y, X) ≥ n + 21 (r + 1),
(cf. (2.1.5)), and by Theorem 1.4 of Chapter I either
T (Y, X) = S(Y, X)
or
dim T (Y, X) = r + n,
so that for r > 0
dim T (Y, X) ≥ n + 12 (r + 1).
1
Thus under our assumptions
m ≥ n+
r < 2(m − n). Hence for
©
ª 2 (r + 1), i.e.
r
r ≥ 2(m−n) (resp. r ≥ max 1, 2(m − n) ) we have Y ∩Dm 6= ∅ (resp. Y r ∩Rm 6=
∅). ¤
2.6. Remark. Corollary 2.5 is nontrivial only if m ≤ 21 (3n − 1). It is clear that
this corollary holds if we only require that X be nonsingular along Y (we considered
the case of nonsingular X in order not to introduce definition of ramification locus
and double point locus in the general situation; cf. e.g. [39; 45; 48]). Example 2.4
shows that the bound in Corollary 2.5 is sharp.
2.7. Remark. If m = n, then from Corollary 2.5 it is easy to deduce that the
linear system |Rn | generated by ramification divisors RL , where L runs through
the set of general (N − n − 1)-dimensional linear subspaces of PN , is ample on X.
This result was already proved in Proposition 2.12 of Chapter I.
Example 2.4 shows that the bound given in Theorem 2.1 is sharp. However in
one important special case, viz. when Y = X, this bound (which takes the form
n ≤ 13 (2N +b−2), where b = dim (Sing X)) can be somewhat improved. This is due
to the fact that for Y = X the subvariety ¡Qz involved
¢ in the proof of Theorem 2.1
can be replaced by the subvariety Yz = p1 ϕ−1(z) , where dim Yz = dim Qz + 1.
2.8. Theorem. Let X n ⊂ PN be a nondegenerate variety, b = dim (Sing X).
Suppose that there exists a point u ∈ PN \X such that the projection π : X → PN −1
with center at u is a J-embedding. Then n ≤ 31 (2N + b) − 1 (i.e. codimPN X ≥
1
2 (n − b + 3)).
Proof. As we already pointed out, the proof is parallel to the proof of Theorem 2.1. In the notations of 2.1, from Proposition 1.10 a) it follows that
L ⊃ TX,x ∀x ∈ Yz \ Sing X,
¡
¢
Yz = p1 ϕ−1 (z) .
(2.8.1)
44
II. PROJECTIONS OF ALGEBRAIC VARIETIES
Let M N −b−1 ⊂ PN be a general linear subspace of codimension b + 1, and let
X 0 = X ∩ M,
Yz0 = Yz ∩ M,
L0 = L ∩ M.
Then the variety X 0 ⊂ PN −b−1 is nonsingular, and from (2.8.1) it follows that
T 0 (Yz0 , X 0 ) = T (Yz0 , X 0 ) ⊂ L0 .
However it is obvious that X 0 6⊂ L0 . Hence
S(Yz0 , X 0 ) 6= T 0 (Yz0 , X 0 ).
(2.8.2)
From (2.8.2) and Theorem 1.4 of Chapter I it follows that
s − b − 1 = dim (SX ∩ M ) ≥ dim SX 0 ≥ dim S(Yz0 , X 0 ) = dim Yz0 + dim X 0 + 1
¡
¢
= (2n + 1 − s) − (b + 1) + (n − b − 1) + 1 = 3n − s − 2b,
i.e.
3n ≤ 2s + b − 1 ≤ 2N + b − 3.
¤
2.9. Remark. In [97] we gave a proof of Theorem 2.8 based on Theorem 1.7 of
Chapter I. In the case when K = C, b = −1 Lazarsfeld showed that Theorem 2.8
can be derived directly from the connectedness theorem of Fulton and Hansen (cf.
[27, § 7]).
2.10. Remark. For n > 1 in the statement of Theorem 2.8 it suffices to assume
that there exists a point u ∈ PN \ X such that the projection π : X → PN −1 with
center in u is J-unramified. In fact, from Proposition 1.5 a) and Theorem 1.4 of
Chapter I it follows that if π is J-unramified, then T 0 X 6= PN and either N ≥
dim SX = 2n + 1, n ≤ 12 (N − 1) ≤ 13 (2N + b) − 1 or SX = T 0 X 3 u and π is a
J-embedding, so that in both cases the conditions of Theorem 2.8 are satisfied.
2.11. Corollary. If a nondegenerate nonsingular variety X n ⊂ PN can be
isomorphically projected to a projective space Pm , m < N , then n ≤ 23 (m − 1). If
n > 1, then it suffices to require existence of unramified projection X → Pm .
2.12. Remark. For m = N −1 the bound given in Corollary 2.11 is sharp, but for
m < N − 1 this bound can be improved (cf. [100] or Corollary 2.16 in Chapter V).
2.13. Remark. The varieties for which the inequality in Theorem 2.8 or Corollary 2.11 turns into equality will be classified in Chapter IV (cf. Chapter IV, Theorems 1.4 and 4.7). We observe that from the proof of Theorem 2.8
¡ it follows that
¢ if
n = 13 (2N + b) − 1, then for a general point z ∈ SX we have dim Yz ∩ Sing X = b,
i.e. Yz contains a component of Sing X.
HARTSHORNE’S CONJECTURE ON LINEAR NORMALITY
45
2.14. Theorem. Let X n ⊂ PN , b = dim (Sing X), n > 31 (2N + b − 1). Then
X is not projection of a variety of the same dimension and degree nontrivially
embedded in a projective space of larger dimension.
Proof. Suppose the converse. Then there exist a variety
0
X 0 ⊂ PN ,
dim X 0 = dim X = n,
deg X 0 = deg X
(2.14.1)
and a linear subspace
0
L ⊂ PN ,
dim L = N 0 − N − 1
0
such that X 0 is nondegenerate and if π : PN 99K PN is the projection with center
L, then π(X 0 ) = X. From our assumptions on the dimension and degree of X 0 it
follows that L ∩ X 0 = ∅.
We may assume that N 0 = N + 1. In fact, if N 0 > N + 1, then we pick a general
linear subspace
L0 ⊂ L,
dim L0 = N 0 − N − 2
0
and denote by π 0 : PN 99K PN +1 the projection with center L0 . Put
X 00 = π 0 (X 0 ),
L00 = π 0 (L),
and let π 00 : PN +1 99K PN be the projection with center L00 (here L00 is a point in
PN +1 ). Then it is clear that L00 ∈
/ X 00 and
π 00 (X 00 ) = π(X 0 ) = X,
dim X 00 = dim X 0 = n,
deg X 00 = deg X 0 = deg X.
Thus
we may
assume that N 0 = N + 1 and L is a point in PN +1 . Let b0 =
¡
¢
0
dim Sing X . Since for L ∈
/ X0
¡
¢
deg X 0 = K(X 0 ) : K(X) · deg X,
from (2.14.1) it follows that π is a finite birational map, so that
¡
¢
π Sing X 0 ⊂ Sing X,
b0 ≤ b.
(2.14.2)
From the condition of the theorem and inequality (2.14.2) it follows that
dim X 0 = n >
2N + b − 1
2(N + 1) + b − 3
2N 0 + b0
=
≥
− 1.
3
3
3
(2.14.3)
In view of Theorem 2.8 and Proposition 1.5 c), from (2.14.3) it follows that
SX 0 = PN
and therefore L ∈ SX 0 .
0
(2.14.4)
46
II. PROJECTIONS OF ALGEBRAIC VARIETIES
Let
ϕ0 : SX 0 → SX 0 ,
p01 : SX 0 → X 0
be the canonical projections. Put
¡
¢
DL = p01 (ϕ0 )−1 (L) .
It is easy to see that
π(DL ) ⊂ Sing X.
(2.14.5)
On the other hand, from (2.14.4) it follows that
¡
¢
dim DL = dim (ϕ0 )−1 (L) ≥ 2n + 1 − N 0 = 2n − N.
(2.14.6)
By our assumption,
2n − N > b + (N − n − 1).
(2.14.7)
Hence from (2.14.5) and (2.14.6) it follows that
¡
¢
¡
¢
b = dim Sing X ≥ dim π(DL ) = dim DL ≥ 2n − N
which contradicts (2.14.7) for N ≥ n + 1. For N = n we have X = PN , and the
assertion of the theorem is obvious. ¤
2.15. Corollary. For n ≥ 23 (N − 1) a nonsingular variety X n ⊂ PN cannot
be obtained by projecting a variety of the same dimension and degree nontrivially
embedded in a projective space of larger dimension.
2.16. Definition. A variety X n ⊂ PN is called linearly normal if the linear
system of
which
¡ of X is complete,
¢
¡ means that the restriction
¡ hyperplane¢ sections
surjective
if X¢¢is nondegenerate, this
map H 0 PN , OPN (1) → H 0 ¡X, OX (1) is
¢
¡
0
0 N
condition is equivalent to H P , OPN (1) ' H X, OX (1) .
Thus a variety X n ⊂ PN is not linearly normal if and only if there exists a
0
0
nondegenerate
variety X 0 ⊂ PN , N 0 > N and a projection π : PN 99K PN such that
¯
∼
0
π ¯X 0 : X → X. The following corollary immediately follows from Corollary 2.15.
2.17. Corollary. For n >
linearly normal.
2
3 (N
− 1) any nonsingular variety X n ⊂ PN is
The simplest case when Corollary 2.17 can be applied and yields a nontrivial
result is the case of threefolds in P5 ; until now it was unknown whether or not they
are linearly normal.
2.18. Remark. Corollary 2.17 was stated as conjecture by R. Hartshorne in 1973
(cf. [33, 4.2]).
2.19. Remark. A variety X n ⊂ PN is called projectively normal if all its Veronese
embeddings
normal
(or, in other words, if the restriction
¡ vk (X), k ¢≥ 1 are linearly
¡
¢
maps H 0 PN , OPN (k) → H 0 X, OX (k) are surjective for all k ≥ 1). Rao [69]
constructed threefolds in P5 which are not projectively normal. We already observed
that all such varieties are linearly normal.
CHAPTER III
VARIETIES OF SMALL CODIMENSION CORRESPONDING
TO ORBITS OF ALGEBRAIC GROUPS
Typeset by AMS-TEX
47
48
VARIETIES OF SMALL CODIMENSION CORRESPONDING TO GROUPS
1. Orbits of algebraic groups, null-forms, and secant varieties
1.1. Let K be an algebraically closed field, char K = 0, and let G be a linear
algebraic group over K acting on a vector space V = K N +1 . Let v ∈ V be a vector
for which Gv is a punctured cone in V , and let
X n = Gv/K ∗ ,→ PN
be the corresponding projective variety.
Let H be the stabilizer of v. By Corollary 2 from [67, no 4], H contains a maximal
unipotent subgroup of G (in our case this simply reflects the fact that each parabolic
subgroup contains a Borel subgroup; cf. [7]). In particular, H contains the unipotent
radical of G, and without loss of generality we may assume that the group G is
reductive (this is a classical theorem of È. Cartan; cf. [12]). Moreover, since we
are interested only in the variety X corresponding to the orbit Gv, we may assume
that G is a semisimple group (perhaps in this situation it would be more natural to
consider groups with one-dimensional center, but the notion of semisimple group is
universally accepted, and it seems inconvenient to use notations in which everything
should be tensored by GL1 ).
Fixing a Borel subgroup B corresponding to a maximal unipotent subgroup
contained in H we can represent v in the form
v = v1 + · · · + vr ,
where for b ∈ B
bvi = Λi (b)vi ,
i = 1, . . . , r,
Λi is the highest weight of the restriction of action of G on an invariant subspace
Vi ⊂ V , and vi is a highest weight vector (primitive element) in Vi . It is clear that
Gv ⊂
r
M
Vi ,
i=1
and without loss of generality we may assume that
V =
r
M
Vi .
i=1
Since X is a projective variety, Gv consists of two orbits, viz. Gv and 0, and therefore
all Λi are collinear (cf. [17], [85]). On the other hand, since Gv is a cone, all Λi lie
in an affine hyperplane (cf. [17]). Thus we may assume that r = 1 and Gv is the
orbit of highest weight vector of an irreducible representation of a semisimple group
G (varieties of such type were considered in [95] and were called HV -varieties). In
particular, from this it follows that the variety X = Gv/K ∗ ⊂ PN is rational
(cf. [72] and 1.3 below) and is defined in PN by quadratic equations (cf. [57]).
1. ORBITS OF ALGEBRAIC GROUPS, NULL–FORMS, AND SECANT VARIETIES 49
Let Λ be the highest weight of our representation, let vΛ be the corresponding
highest weight vector, and let g be the Lie algebra of the group G. It is clear that
the variety of tangents
[
TX =
TX,x
x∈X
corresponds to the affine cone
GgvΛ ⊂ V.
(1.1.1)
Furthermore, if PΛ ⊂ G is the parabolic subgroup stabilizing the line KvΛ (or,
which is the same, the point xΛ ∈ X corresponding to vΛ ), then the stabilizer of
gvΛ in G coincides with PΛ (this follows from Corollary 2.8 in Chapter I).
Similarly, let NΛ ⊂ V ∗ be the subspace of points corresponding to hyperplanes
passing through gvΛ (i. e. the ‘normal’ subspace). Then the variety corresponding
to the cone GNΛ coincides with the dual variety X ∗ ⊂ (PN )∗ (here we consider the
contragredient representation of G in V ∗ ). Moreover, the stabilizer of NΛ coincides
with that of gvΛ . It is well known that from this it follows that the varieties T X
and X ∗ (as well as X) are rational and arithmetically Cohen-Macaulay (cf. [47]).
We proceed with finding out which of the varieties corresponding to orbits of
highest weight vectors are complete intersections. If X is a complete intersection,
then, according to Proposition 2.10 of Chapter I,
¡ ¡
¢¢
X ∗ = γN −1 P NPN /X (−1) ,
where NPN /X is the normal bundle and
¡
¢
γN −1 : P NPN /X (−1) → X ∗
is a finite birational morphism. Since, as we have already observed, in our case
∗
N ∗
the variety X ∗ is normal, from this it follows
¡ that X ⊂¢ (P ) is a nonsingular
∗
hypersurface. On the other hand, X ' P NPN /X (−1) is a projective bundle
over X with fiber PN −n−1 . Hence N − n − 1 ≤ 0, i. e. either X = Pn or X is
a hypersurface. In the last case it is clear that X is a quadric. Summing up, we
obtain the following result.
1.2. Theorem. Let X n = Gv/K ∗ ⊂ (V \0)/K ∗ = PN be the projective variety
corresponding to an orbit Gv ⊂ V of an irreducible representation of an algebraic
group G which is a punctured cone. Then X = Gx, where x is the point of PN
corresponding to a highest weight vector v. Furthermore, X, T X, and X ∗ are
rational arithmetically Cohen-Macaulay varieties. The variety X is defined in PN
by quadratic equations. Moreover, either X = Pn (i. e. G acts transitively on V \ 0)
or X is a nonsingular quadric, or X is not a complete intersection.
By analogy with [95], we call projective varieties satisfying the conditions of
Theorem 1.2 HV -varieties.
1.3. We turn to secant varieties. In the above notations, let M be the lowest
weight of the representation of G in V , and let vM be the corresponding weight
vector. Although the orbit GvΛ does not necessarily contain all weight vectors, we
have
vM ∈ GvΛ
50
VARIETIES OF SMALL CODIMENSION CORRESPONDING TO GROUPS
since
M = w0 (Λ),
where w0 is the involution in the Weyl group W of the group G transforming the
positive Weyl chamber to the negative one (cf. [9, ch. VI, § 1, no 6, Corollary 3]),
and we may assume that w0 is contained in the normalizer of a maximal torus of
G. Let xΛ (resp. xM ) be the point in X corresponding to vΛ (resp. vM ), and let
PΛ (resp. PM ) be the stabilizer of xΛ (resp. xM ). Consider the orbit of the point
xΛ ×xM ∈ X × X under the natural action of G on X × X. It is clear that the
stabilizer of xΛ ×xM coincides with PΛ ∩ PM . Since PΛ contains the ‘upper’ and PM
the ‘lower’ Borel subgroup of G, we have
dim (PΛ · PM ) = dim G
(cf. [7, ch. IV, Theorem 14.1]). Hence
¡
¢
dim G · (xΛ ×xM ) = dim G − dim (PΛ ∩ PM ) =
(dim G − dim PΛ ) + (dim PΛ − dim (PΛ ∩ PM )) =
dim X + (dim (PΛ · PM ) − dim PM ) = 2 dim X = 2n
¡
±
¢
(similar computation shows that dim BM BM ∩ PΛ = n, so that BM · xΛ = X
and from [72] it follows that X is a rational variety). Hence the orbit G · (xΛ ×xM )
is dense in X × X and
SX = G hxΛ , xM i.
(1.3.1)
Let U ⊂ V be the plane spanned by the vectors vΛ and vM , let N be the cone of
null-forms in V (i. e. N is the subset in V defined by vanishing of all G-invariant
polynomials), and let Z ⊂ PN be the projective variety corresponding to the cone
N. It is clear that X ⊂ Z. Consider the action of the maximal torus T ⊂ B on U .
Let
v = αΛ vΛ + αM vM ,
where αΛ , αM 6= 0. Then
Tv ⊂ U
and there are two possibilities: either
Λ + M 6= 0,
dim T v = 2
Λ + M = 0,
dim T v = 1.
or
In the first case
Gv ⊃ T v 3 0
and therefore
GU = Gv ⊂ N,
SX ⊂ Z
(1.3.2)
(cf. [17], [85]). In the second case
GU = G(Kv);
(1.3.3)
1. ORBITS OF ALGEBRAIC GROUPS, NULL–FORMS, AND SECANT VARIETIES 51
examples show that in this case SX may lie or not lie in Z.
We observe that from (1.3.2) and (1.3.3) it follows that if
z ∈ hxΛ , xM i ,
z 6= xΛ , xM ,
then
Gz = SX.
The involution w0 for simple Lie groups is described in the tables in [9]. In
particular, w0 = −1 (and therefore Λ + M = 0 for all representations) if and only
if G is a simple group of one of the following types: A1 , Br (r ≥ 2), Cr (r ≥ 2),
D2l (l ≥ 2), E7 , E8 , F4 , G2 .
From (1.3.2), (1.3.3), and (1.3.4) it follows that if SX = PN , then either N = V ,
i. e. IG [V ] = K, where IG [V ] is the algebra of polynomials on V invariant with
respect to the action of G, or Gv is a hypersurface in V and, since
N ⊃ Gv,
dim N ≥ dim Gv,
IG [V ] = K[F ], where F generates the ideal of N in K[V ]. All representations
for which the algebra of invariants has such form have been classified (cf. [46], [85
(addendum)]), viz. if G is a simple group, then IG [V ] = K if and only if
G = SLr , Sp2r , ∧2 (SL2r+1 ) (r > 1), Spin10 ;
(1.3.5)
IG [V ] = K[F ] if and only if
G =SOr (r 6= 4), ∧2 (SL2r ) , S 2 (SL2r ) (r ≥ 1), ∧3 (SLr ) (r = 6, 7, 8),
∧30 (Sp6 ) , S 3 (SL2 ) , Spinr (r = 7, 9, 11, 12, 14), E6 , E7 , G2 ,
(1.3.6)
where ∧ denotes exterior power of representation and ∧0 is the ‘principal part’ of
decomposition of ∧ into irreducible summands. From the lists (1.3.5) and (1.3.6) it
is easy to select representations for which SX = PN . It turns out that if SX = PN
for a variety X n ⊂ PN corresponding to representation of a simple group G, then
there are the following possibilities:
X = Pn , X = Qn ⊂ Pn+1 , X = v3 (P1 ) ⊂ P3 ,
X = G(4, 1)6 ⊂ P9 , X = G(5, 2)9 ⊂ P19 , X = S 10 ⊂ P15 ,
X = S 15 ⊂ P31 , X = C 6 ⊂ P13 , X = E 27 ⊂ P55 ,
where Qn is a nonsingular quadric, v3 (P1 ) is a rational cubic curve, S 10 and S 15
are the spinor varieties parametrizing linear subspaces on a nonsingular eight- and
ten-dimensional quadric respectively (cf. [11], [35], [87]), C 6 is the variety corresponding to the orbit of highest weight vector of representation ∧30 (Sp6 ), and E 27
is the variety corresponding to the orbit of highest weight vector of the standard
representation of the group E7 .
Summing up, we obtain the following result.
52
VARIETIES OF SMALL CODIMENSION CORRESPONDING TO GROUPS
1.4. Theorem. If X = GxΛ = GxM , then SX = GhxΛ , xM i. If Λ + M 6= 0,
then SX ⊂ Z and in particular SX 6= PN if the representation of G in V has at
least one nontrivial invariant. If SX = PN , then IG [V ] = K[F ] (here F is a form
which may belong to K). Furthermore, if G is a simple group, then X is one of the
following nine varieties: Pn , Qn , v3 (P1 ), G(4, 1), G(5, 2), S 10 , S 15 , C 6 , E 27 .
1.5. By Theorem 1.2 we may assume that
G = G1 × · · · × Gd ,
V = V1 ⊗ · · · ⊗ Vd ,
d ≥ 1,
where Gi (i = 1, . . . , d) is a simple group and representation G → Aut V is a tensor
product of nontrivial irreducible representations Gi → Aut Vi with highest weights
Λi and highest weight vectors vi ∈ Vi . It is clear that the highest weight Λ of the
representation G → Aut V is equal to Λ1 + · · · + Λd and the corresponding highest
weight vector can be represented in the form v = v1 ⊗ . . . ⊗vd .
Let Xi ⊂ PNi = P(Vi ) be the projective variety corresponding to the orbit
Gi vi , and let ni = dim Xi , (i = 1, . . . , d). Then it is clear that the variety X n ⊂
PN = P(V ) corresponding to the orbit Gv is projectively isomorphic to the Segre
¡
¢n1 +···+nd
embedding of X1 × · · · × Xd
in P(N1 +1)...(Nd +1)−1 .
In what follows we shall need information about secant varieties of projective
varieties corresponding to orbits of highest weight vectors of irreducible representations of semisimple (but not simple) algebraic groups. In view of 1.5, it suffices to
prove the following general theorem in whose statement we no longer assume that
varieties correspond to group actions.
1.6. Theorem. Let {X n ⊂ PN } = {X1n1 × · · · × Xdnd ⊂ P(N1 +1)...(Nd +1)−1 },
where Xini ⊂ PNi , 0 < n1 ≤ · · · ≤ nd , d ≥ 2 (the Segre embedding). Then
dim SX = 2n + 1 except in the case when d = 2, X1 = Pn1 , X2 = Pn2 ; in this
last case dim SX = 2n − 1. Furthermore, SX = PN if and only if {X n ⊂ PN } =
{P1 × Pn−1 ⊂ P2n−1 } or {X n ⊂ PN } = {P1 × V n−1 ⊂ P2n+1 }, where n ≥ 2 and
V n−1 ⊂ Pn is a hypersurface (we observe that the variety P1 ×P1 ×P1 ⊂ P7 belongs
to this last case).
Proof. Arguing by induction, it is easy to reduce everything to the case d = 2.
Let x = (x1 , x2 ); x0 = (x01 , x02 ) be a general pair of points of the variety
X = X1 × X2 ⊂ PN1 × PN2 = P,
and let z be a general point of the chord hx, x0 i. If dim SX ≤ 2n, then
dim Yz ≥ 1,
(1.6.1)
where, in the notations of Section 1 of Chapter I and Theorem 2.8 of Chapter II,
¯
©
ª
¯
Yz = (pX )1 (ϕ−1
X (z)) = x ∈ X ∃y ∈ X, y 6= x, z ∈ hx, yi .
On the other hand, it is clear that
Yz ⊂ Ỹz ,
(1.6.2)
1. ORBITS OF ALGEBRAIC GROUPS, NULL–FORMS, AND SECANT VARIETIES 53
where
¯
ª
©
¯
Ỹz = (pP )1 (ϕ−1
P (z)) = x ∈ P ∃y ∈ P, y 6= x, z ∈ hx, yi .
(1.6.3)
But it is well known (cf. e. g. [33; 38]) that if x01 6= x1 , x02 6= x2 , then
Ỹz = hx1 ×x01 i × hx2 , x02 i
(1.6.4)
is a nonsingular two-dimensional quadric and
¯
©
ª
Yz ⊂ Ỹz ∩ X = (y1 , y2 ) ¯ y1 ∈ hx1 , x01 i ∩ X1 , y2 ∈ hx2 , x02 i ∩ X2 .
(1.6.5)
Suppose that at least one of the varieties X1 , X2 , say X1 is not a projective space.
Then from (1.6.5) it follows that Yz consists of a finite number of points and a
finite number of lines from one of the two families of lines on Ỹz . But from (1.6.2),
(1.6.3), and (1.6.4) it follows that, along with each line l ⊂ Yz ⊂ Ỹz from one family
of lines on Ỹz , Yz contains
a ¢line from the other family, viz. the line intersecting l
¡
at the point l ∩ (pP )1 ψP−1 (z) . In view of (1.6.1) we come to a contradiction. Thus
if dim SX < 2n + 1, then X n = Pn1 × Pn2 is a Segre variety.
Suppose now that SX = PN . Then
N = (N1 + 1) . . . (Nd + 1) − 1 ≤ 2n + 1 = 2(n1 + · · · + nd ) + 1,
(1.6.6)
and it is clear that d = 3, n1 = n2 = n3 = N1 = N2 = N3 = 1 or d = 2. In the last
case from (1.6.6) it follows that there are three possibilities: n1 = n2 = N1 = N2 =
2; n1 = N1 = 2, n2 = N2 = 3; n1 = N1 = 1, N2 = n2 + 1. ¤
1.7. Corollary. In the conditions of Theorem 1.2 suppose that the semisimple
group G is not simple. Then dim SX ≤ 2n if and only if X n ⊂ PN = Pn1 × Pn2 ⊂
Pn1 n2 +n1 +n2 (in which case dim SX = 2n − 1). Furthermore, SX = PN if and only
if {X n ⊂ PN } = {P1 × Pn−1 ⊂ P2n−1 } or {X n ⊂ PN } = {P1 × Qn−1 ⊂ P2n+1 },
where Qn−1 ⊂ Pn is a nonsingular quadric.
Proof. Corollary 1.7 immediately follows from Theorems 1.6 and 1.2.
¤
54
VARIETIES OF SMALL CODIMENSION CORRESPONDING TO GROUPS
2. HV -varieties of small codimension
In this section we study those varieties X n ⊂ PN corresponding to orbits of
highest weight vectors of irreducible representations of semisimple algebraic groups
for which N ≤ 2n + 1.
2.1. Proposition. If X n = GxΛ ⊂ PN = P(V ), where G → Aut V is an
irreducible representation of a semisimple, but not simple group G such that N ≤
2n + 1, then X is of one of the following types:
a)
b)
c)
d)
X n = P1 × Pn−1 ⊂ P2n−1 (n ≥ 2);
X n = P1 × Qn−1 ⊂ P 2n+1 , where Qn−1 is a nonsingular quadric (n ≥ 2);
X 4 ⊂ P2 × P2 ⊂ P8 ;
X 5 = P2 × P3 ⊂ P11 .
Furthermore, in cases a) and b) SX = PN and in cases c) and d) SX 6= PN .
Proof. Proposition 2.1 immediately follows from Corollary 1.7. ¤
2.2. Proposition. If X n = GxΛ ⊂ PN = P(V ), where G → Aut V is an
irreducible representation of a simple group G such that N ≤ 2n + 1, then dim V <
dim G except in the following cases:
a) rk G = 2, dim G = dim V , G → Aut V is the adjoint representation;
b) G = SL2 , G → Aut V is the adjoint representation, X = v2 (P1 ) = Q1 ⊂ P2
is a conic;
c) G = SL2 , X = v3 (P1 ) ⊂ P3 is a rational cubic curve.
Proof. Since the parabolic subgroup PΛ stabilizing the point xΛ contains a Borel
subgroup BΛ ⊂ G, we have
n = dim X = dim G − dim PΛ ≤ dim G − dim BΛ = 21 (dim G − rk G),
i. e.
dim G ≥ 2n + rk G.
(2.2.1)
From (2.2.1) it follows that for N ≤ 2n + 1
dim G ≥ 2n + rk G ≥ dim V + (rk G − 2).
(2.2.2)
The inequality (2.2.2) shows that if rk G > 2, then dim V < dim G. The proof of
Proposition 2.2 is completed by a direct check. ¤
All irreducible representations G → Aut V of simple algebraic groups G for
which dim V < dim G were classified in [2] and [20]. Using tables from these
papers (which for our purposes should be complemented by adding the adjoint
representations of the groups SL3 , Sp4 , and G2 and the second and third symmetric
powers of the standard representation of SL2 ) we select those representations for
which N ≤ 2n + 1.
First we describe those varieties X n ⊂ PN for which SX 6= PN . Special attention
will be devoted to the case of Severi varieties which is important for what follows.
2. HV –VARIETIES OF SMALL CODIMENSION
55
2.3. Definition. A nonsingular nondegenerate (not necessarily homogeneous)
variety X n ⊂ PN is called Severi variety if n = 23 (N − 2) and SX 6= PN .
We recall that from Corollary 2.11 in Chapter II it follows that for n > 23 (N − 2)
we have SX = PN , so that for a fixed N Severi varieties have maximal dimension
among the varieties which can be isomorphically projected to a projective space of
smaller dimension. Complete classification of Severi varieties will be given in Chapter IV, and here we restrict ourselves to classifying homogeneous Severi varieties (a
posteriori all Severi varieties turn out to be homogeneous).
First we consider the case when the group G is not simple.
2.4. Theorem. If X n ⊂ PN is a projective variety corresponding to the orbit
of highest weight vector of an irreducible representation of a simple group G in a
vector space V N +1 , where N ≤ 2n + 1 and SX 6= PN , then either G is a simple
group or X is a Segre variety of the form Pn1 × Pn2 ⊂ Pn1 n2 +n1 +n2 , where n1 = 2,
2 ≤ n2 ≤ 3. In the last case X is a Severi variety if and only if n1 = n2 = 2,
N = n1 n2 + n1 + n2 = 8; for this Severi variety V can be identified with the space
M3 of 3×3-matrices over the field K, X corresponds to the cone of matrices of rank
less than or equal to one, and SX is the cubic hypersurface corresponding to the
cone of degenerate matrices defined by equation det M = 0, M ∈ M3 . Furthermore,
for the above variety X = Sing SX, (SX)∗ ' X, X ∗ ' SX.
Proof. The first assertion of the theorem follows from Proposition 2.1. Furthermore, it is clear that the variety Pn1 × Pn2 corresponds to the standard representation of the group SLn1 +1 ×SLn2 +1 in the space of matrices of order (n1 +1)×(n2 +1),
and the orbit of highest weight vector consists of matrices of rank one. In particular,
for the Severi variety P2 × P2 we have
V ' M3 ,
ISL3 ×SL3 [V ] = K[det].
By Theorem 1.4, in this case
SX ⊂ Z ' (N \ 0)/K ∗ .
Hence SX = Z is a cubic hypersurface in P8 (corresponding to the cone of matrices
of rank less than or equal to 2), X = Sing SX, and from the structure of orbits of
the representation of G in V and the contragredient representation of G in V ∗ it
immediately follows that there exist natural G-isomorphisms
(SX)∗ ' X,
X ∗ ' SX.
¤
2.5. Now let G be a simple group. Proposition 2.2 and analysis of the results of
[41] and tables from [2] and [20] yield the following list of irreducible representations
for which dim SX < N ≤ 2n + 1 (in the statement of our results we denote by ϕi
the i-th fundamental weight in the notations of [9] and [94]):
56
VARIETIES OF SMALL CODIMENSION CORRESPONDING TO GROUPS
A1 ) G = SL3 , Λ = 2ϕ1 (or Λ = 2ϕ2 ), n = 2, N = 5, X = v2 (P2 ) is the
Veronese surface. The space P5 is identified with projectivization of the vector
space of symmetric 3 × 3-matrices, and the algebra of invariants has the form
IG [V ] = K[det], so that Z is a cubic hypersurface in P5 . From Theorem 1.4 it follows
that SX ⊂ Z, and therefore SX = Z, so that X is a Severi variety. The surface
X corresponds to the cone of symmetric matrices of rank less than or equal to one,
and the variety SX corresponds to the cone of degenerate symmetric matrices.
Moreover, X = Sing SX, and from the structure of orbits of the representation of
G in V and the contragredient representation of G in V ∗ it immediately follows
that there exist natural G-isomorphisms (SX)∗ ' X, X ∗ ' SX. The variety X ∗
is normal, and for α ∈ Sing X ∗ ' X the hyperplane Lα is tangent to X along a
nonsingular conic Q1 ⊂ X.
A2 ) G = SL3 , Λ = ϕ1 = ϕ2 (adjoint representation), n = 3, N = 7. The
space V is identified with the vector space of 3 × 3-matrices with trace zero, and
the group G = SL3 is identified with a subgroup of the group SL3 × SL3 acting
in the space of all 3 × 3-matrices as tensor product of the representations of SL3
corresponding to ϕ1 and ϕ2 . It is easy to see that the orbit of highest weight vector
of our representation is the intersection of the orbit of highest weight vector of
this representation of SL3 × SL3 with the hyperplane of matrices with trace zero.
Hence X is a nonsingular hyperplane section of the Segre variety P2 × P2 ⊂ P8 or,
equivalently, X is the projectivization of tangent bundle of P2 naturally embedded
in P7 . The algebra of invariants has the form IG [V ] = K[Q, det], where Q is the
quadratic form defined by trace. In this case Λ + M = 0, Z is a five-dimensional
variety, SX is a cubic hypersurface in P7 , and Sing SX = X. Here representation
G → Aut V is naturally isomorphic to the contragredient representation G →
Aut V ∗ , and the dual variety X ∗ ⊂ P7∗ is defined by equation 4Q3 − 27 det2 = 0.
Furthermore, X ∗ ⊃ π(P2 × P2 ), where π : P8 99K P7 , P8 = P(M3 ) is the projection
with center at the point corresponding to the unit matrix and the subvariety P2 ×
P2 ⊂ P8 corresponds to matrices of rank one, Z ∩ π(P2 × P2 ) = X, and Sing X ∗ =
Z ∪ π(P2 × P2 ).
A3 ) G = SL6 , Λ = ϕ2 (or Λ = ϕ4 ), n = 8, N = 14, X = G(5, 1) is the
Grassmann variety of lines in P5 . The space V is identified with the vector space
of skew-symmetric matrices of order 6 × 6, and the algebra of invariants has the
form IG [V ] = K[Pf], where Pf denotes the Pfaffian of skew-symmetric matrix. The
cone of null-forms consists of three orbits, viz. {0}, the set of nonzero decomposable
bivectors corresponding to X, and the set of bivectors of rank four. Theorem 1.4
shows that SX ⊂ Z and therefore SX = Z. Thus SX is a cubic hypersurface
in P14 , X is a Severi variety, Sing SX = X, and from the structure of orbits
of the representation of G in V and the contragredient representation of G in
V ∗ it immediately follows that there exist natural G-isomorphisms (SX)∗ ' X,
X ∗ ' SX. Moreover, X ∗ is a normal variety, and for α ∈ Sing X ∗ ' X the
hyperplane Lα is tangent to X along a nonsingular quadric Q4α ⊂ X.
A4 ) G = SL7 , Λ = ϕ2 (or Λ = ϕ5 ), n = 10, N = 20, X = G(6, 1) is the
Grassmann variety of lines in P6 . This example is similar to the previous one. Here
N = V and V consists of four orbits, viz. {0}, the set of bivectors of rank two
corresponding to X, the set of bivectors of rank four, and the set of bivectors of
2. HV –VARIETIES OF SMALL CODIMENSION
57
rank six which is dense in V . The variety SX corresponds to the cone of bivectors of
rank less than or equal to four, dim SX = 2n − 3 = 17, Sing SX = X, (SX)∗ ' X,
X ∗ ' SX, the variety X ∗ is normal, and for α ∈ Sing X ∗ ' X the hyperplane
Lα is tangent to X along a six-dimensional Grassmannian G(4, 1) while for α ∈
X ∗ \ Sing X ∗ this hyperplane is tangent to X along a projective plane.
C) G = Sp6 , Λ = ϕ2 , n = 7, N = 13. We consider Sp6 as subgroup of SL6
and restrict the representation of SL6 in the vector space of skew-symmetric 6 × 6matrices described in A3 ) on Sp6 . The resulting representation of Sp6 is a direct
sum of our representation and a trivial representation
¶ in the one-dimensional space
µ
0
13
spanned by the skew-symmetric matrix
. It is easy to see that this case
−13 0
is similar to A2 ), viz. X is a hyperplane section of the variety G(5, 1) from example
A3 ). The algebra of invariants is free and is generated by two elements Q and Pf of
degrees two and three respectively (Q and Pf are coefficients of the ‘characteristic
pfaffian polynomial’). In this case Λ + M = 0, Z is an eleven-dimensional variety,
SX is a cubic hypersurface in P13 and Sing SX = X. Representation G → Aut V
is naturally isomorphic to the contragredient representation G → Aut V ∗ , and the
dual variety
X ∗¢⊂ (P13 )∗ is defined by equation 4Q3 − 27 Pf 2 = 0. Furthermore,
¡
X ∗ ⊃ π G(5, 1) , where π: Pµ14 99K P13¶is the projection with center at the point
0
13
corresponding to the matrix
, the subvariety G(5, 1) ⊂ P14 corresponds
−13 0
to decomposable bivectors,
¡
¢
Z ∩ π G(5, 1) = X,
¡
¢
Sing X ∗ = Z ∪ π G(5, 1) .
E) G = E6 , Λ = ϕ1 (or Λ = ϕ6 ), n = 16, N = 26. The space V is identified
with the 27-dimensional exceptional Jordan algebra J3 of Hermitean 3 × 3-matrices
over the Cayley algebra (multiplication in the commutative, but not associative
algebra J3 is defined as follows: v ◦w = 21 (vw + wv), where v, w ∈ J3 and vw and
wv are the ordinary products of matrices; cf. [43]). Trace defines a quadratic form
Q(v) = Tr (v ◦v) = Tr (v 2 ) and determinant a cubic form det v on the space V (cf.
[23]). It is known (cf. [13; 23; 44]) that the group E6 is identified with the group
of linear transformations of V preserving det. Thus IG [V ] = K[det]. The cone
N is a union of three orbits, viz. {0}, the punctured cone over X (consisting of
the so-called ‘primitive idempotents’; cf. [23]), and the complement of this cone.
From Theorem 1.4 it follows that SX = Z is a cubic hypersurface in P26 , so
that X is a Severi variety, Sing SX = X, and from the structure of orbits of the
representation of G in V and the contragredient representation of G in V ∗ it follows
that there exist natural G-isomorphisms (SX)∗ ' X, X ∗ ' SX. Furthermore, X ∗
is a normal variety, and for α ∈ Sing X ∗ ' X the hyperplane Lα is tangent to X
along a nonsingular quadric Q8α ⊂ X. The above interpretation of representation
E6 → Aut V is essentially due to Chevalley, Schafer, Freudenthal, Springer, and
Jacobson (cf. [13; 23; 44; 76]). However the representation itself was studied already
in È. Cartan’s dissertation published in 1894 (cf. [12], ch. VIII, § 8, no 5). Cartan
has also written out defining equations for SX and X. The fact that X is a Severi
variety was discovered by Chevalley [13] and independently by Lazarsfeld [56] who
used Kempf’s results [47].
58
VARIETIES OF SMALL CODIMENSION CORRESPONDING TO GROUPS
F) G = F4 , Λ = ϕ4 , n = 15, N = 25. The group F4 is identified with a subgroup
of the group E6 acting on the space J3 as described in E). More precisely, F4 is
the subgroup of E6 consisting of linear transformations preserving the unit element
e ∈ J3 , so that F4 coincides with the group of automorphisms Aut J3 of the algebra
J3 . The representation of F4 in J3 splits into a direct sum of two representations,
viz. the trivial representation in the one-dimensional space spanned by e and an
irreducible representation in the subspace V = e⊥ ⊂ J3 (the orthogonal complement
with respect to the form Q). It is clear that V coincides with the hyperplane
of matrices with trace zero in J3 and X and SX are identified with hyperplane
sections of the corresponding varieties from E). The algebra of invariants has the
form IG [V ] = K[Q, det]. Thus the case F) is similar to the cases A2 ) and C) (in
particular, as in these cases, all elements x ∈ e⊥ = V satisfy characteristic equation
x3 − Q(x) · x − det x · e = 0 (cf. [23]), Λ + M = 0, Z is a variety of dimension 23,
SX is a cubic hypersurface in P25 , Sing SX = X). The representation G → Aut V
is naturally isomorphic to the contragredient representation G → Aut V ∗ , and the
dual variety X ∗ ⊂ P25∗ is defined by equation 4Q3 − 27 det2 = 0. Moreover, X ∗ ⊃
π(E), where π : P26 99K P25 is the projection with center at the point corresponding
to K · e and E is the variety described in E),
Z ∩ π(E) = X,
Sing X ∗ = Z ∩ π(E).
The 25-dimensional representation of the group F4 was also described by È. Cartan
(cf. [12, ch. VIII, § 8, no 8]), but it is not clear how to deduce the connection between
the simplest nontrivial representations of the groups F4 and E6 from his description.
2.6. Remark. As we have already observed, the varieties described in A2 ), C),
and F) are hyperplane sections of the Severi varieties P2 × P2 ⊂ P8 , G(5, 1) ⊂ P14 ,
and E 16 ⊂ P26 respectively. The (nonsingular) hyperplane sections of the fourth
Severi variety, viz. the Veronese surface from example A1 ), are curves in P4 , and
they no longer satisfy the condition N ≤ 2n + 1. However they admit a similar
description.
A0 ) G = SL2 , Λ = 4ϕ1 , n = 1, N = 4, X = v4 (P1 ) is a linearly normal curve
in P4 . The space V is identified with the vector space of symmetric 3 × 3-matrices
with trace zero. The group G is identified with the subgroup of orthogonal matrices
SO3 ⊂ SL3 whose representation in the six-dimensional vector space of symmetric
matrices splits into a direct 
sum of two
 representations, viz. the trivial represen1 0 0
tation in the subspace K ·  0 1 0  and our representation in the subspace

⊥
0 0 1
1 0 0


V =K· 0 1 0
(scalar product with respect to the bilinear form defined
0 0 1
by trace). Thus X is the hyperplane section of the Veronese surface v2 (P2 ) ⊂ P5
(cf. A1 )) corresponding to the hyperplane of matrices with trace zero. The algebra of invariants has the form IG [V ] = K[Q, det], where Q is the quadratic form
defined by trace. In this case Λ + M = 0, Z is a surface, SX is a cubic hypersurface in P4 , and Sing SX = X. The representation G → Aut V is naturally
isomorphic to the contragredient representation G → Aut V ∗ , and the dual variety
2. HV –VARIETIES OF SMALL CODIMENSION
59
¡
¢
X ∗ ⊂ P4∗ is defined by equation 4Q3 − 27 det2 = 0. Moreover, X ∗ ⊃ π v2 (P2 ) ,
where π : P5 99K P4 is the projection with center at the point corresponding to the
2
unit matrix,
the
to¢ symmetric matrices with trace zero,
¡
¢ surface v2 (P∗ ) corresponds
¡
2
Z ∩ π v2 (P ) = X, Sing X = Z ∪ π v2 (P2 ) . Thus example A0 ) is quite similar
to examples A2 ), C), and F).
2.7. Remark. In all cases except A0 ), A2 , C), and F) we gave a complete description of orbits of the representation of G in V . Now we describe the orbits in
the remaining cases. Put
Vab = (Q, d)−1 (a, b),
where a, b ∈ K, d = det in cases A0 ), A2 ), and F) and d = Pf in case C) (so that
in particular V00 = N), and let
D(a, b) = 4a3 − 27b2
be the discriminant of characteristic polynomial in cases A0 ), A2 ), and F) and the
discriminant of characteristic pfaffian polynomial in case C). Then Vab is an orbit
of dimension 2rk G−1 · 3 if D(a, b) 6= 0, and Vab consists of three orbits of dimensions
0, 2rk G , and 2rk G−1 · 3 if D(a, b) = 0, ab 6= 0.
Summing up, we obtain the following result.
2.8. Theorem. If a variety X n ⊂ PN corresponds to the orbit of highest weight
vector of an irreducible representation of a simple Lie group G and dim SX < N ≤
2n + 1, then X is one of the seven varieties A1 )–A4 ), C), E), F). The varieties
A1 ), A3 ), and E) are Severi varieties, and the varieties A0 ), A2 ), C), and F) are
hyperplane sections of Severi varieties.
Combining Theorems 2.4 and 2.8 we obtain the following result.
2.9. Theorem. Over an algebraically closed field K of characteristic zero there
exist exactly four Severi varieties corresponding to orbits of linear actions of algebraic groups, viz.
1)
2)
3)
4)
v2 (P2 ) ⊂ P5 (A1 ) );
P2 × P2 ⊂ P8 ;
G(5, 1)8 ⊂ P14 (A3 ) );
E 16 ⊂ P26 ( E) ).
If X is any of these varieties, then the secant variety SX is a cubic hypersurface
corresponding to the cone of null-forms N. Furthermore, N consists of two orbits,
so that Sing SX = X. There exist natural equivariant isomorphisms X ∗ ' SX,
(SX)∗ ' X. Nonsingular hyperplane sections of these Severi varieties also correspond to orbits of linear actions of algebraic groups.
Various geometric characteristics of the projective variety corresponding to the
orbit of highest weight vector of an irreducible representation of a semisimple Lie
group can be computed in terms of the corresponding representation. For example,
results of § 24 of [8] allow to compute the Hilbert polynomial (and in particular
the degree), and results of §§ 1.4–1.6 of [3] show how to construct triangulation
(and in particular allow to compute the Betti numbers). Let HX (t) be the Hilbert
60
VARIETIES OF SMALL CODIMENSION CORRESPONDING TO GROUPS
polynomial, let br (X) be the r-th Betti number (from [3] it follows that br (X) = 0
for all odd r), and let e(X) be the Euler characteristic of a variety X. One can derive
universal formulae for all these invariants which are valid for all Severi varieties
from Theorem 2.9. However since the Veronese surface is rather special, it is more
convenient to give the corresponding formulae separately for this surface and the
other Severi varieties.
2.10.
Proposition.
For the Veronese surface we have H(t) = (t + 1)(2t + 1),
¡
¢
deg v2 (P2 ) = 4, b0 = b2 = b4 = 1, e = 3. For the homogeneous Severi variety X n ,
n > 2 we have
3
¶nk
n−1 µ
4Y
t
H(t) =
+1
,
k
k=1
where nk = 1 for k <
n
4
≤ k ≤ n2 ,
¡n
¢
3n
1 ! n!
···n
4 −
¡
¢
¡
¢
deg X n = n4
=
n
n
3n
4 ··· 2
2 !
4 −1 !
and k >
n
2,
nk = 2 for
n
4
(deg X 4 = 6, deg X 8 = 14, deg X 16 = 78), b2k = b2n−2k = nk for 1 ≤ k < n2 ,
b0 = b2n = 1, bn = 3, and the Euler characteristic is equal to the number of
quadratic generators of the ideal of X, i. e. e(X) = 3n
2 + 3 = N + 1.
2.11. Let X n ⊂ PN = P(V ) be the variety corresponding to the orbit of highest
weight vector vΛ ∈ V , with respect to the action of G on V , let xΛ and xM be the
points of X corresponding respectively to the highest and lowest weight vectors,
and let z be a general point of the line hxΛ , xM i. Denote by Sz the stabilizer of the
point z. Then
dim (Sz · xΛ ) = dim (Sz /PΛ ∩ Sz ) = 2n + 1 − dim Gz = 2n + 1 − s,
where s = dim SX (cf. 1.3). In particular, for the Severi varieties described in
Theorem 2.9 Sz · xΛ is a n/2-dimensional quadric. Shifting this quadric by means
of the group G we obtain an n-dimensional family of quadrics on the Severi variety
X n.
The family of quadrics on Severi varieties can be also obtained in the following
way. Besides the parabolic subgroup PΛ , the Borel subgroup B = BΛ is also
contained in the parabolic subgroup P−M corresponding to highest weight vector
of the contragredient representation (the corresponding highest weight is equal to
−M). Since for Severi varieties Λ + M 6= 0, it is clear that P−M 6= PΛ , but
C/P−M ' G/PΛ ,
which yields another interpretation of the isomorphism (SX)∗ ' X. This situation
was essentially described by È. Cartan (cf. [12, ch. VIII, § 8, no 11]). Consider the
orbit of the point xΛ with respect to the action of the subgroup P−M . Denote by
H−M the semisimple group corresponding to P−M (the Coxeter graph for H−M is
obtained from the Coxeter graph for G by deleting the vertices at which the weight
M is distinct from zero). It is clear that
P−M · xΛ = H−M · xΛ ,
2. HV –VARIETIES OF SMALL CODIMENSION
61
and the action of H−M corresponds to the restriction of the weight Λ on the maximal
torus of H−M . For the Severi varieties 1)–4) from Theorem 2.9 this representation
of H−M has the following form:
2
1)
◦
2)
◦ ◦
3)
◦—◦—◦ ◦
◦
1
◦—◦—◦
◦
4)
1
(SL2 , 2ϕ1 );
1
1
(SL2 × SL2 , ϕ1 ⊕ ϕ1 );
(SL4 × SL2 , ϕ2 ⊕ 0);
(Spin10 , ϕ1 ) .
In all these cases QΛ = H−M · xΛ is a nonsingular quadric of dimension 21 n.
Shifting QΛ by means of the group G we obtain the desired family of quadrics.
2.5 and 2.11 yield the following proposition.
2.12. Proposition. On each of the Severi varieties X n ⊂ PN described in
Theorem 2.9 there exists an n-dimensional family of 21 n-dimensional quadrics. The
quadrics on X n are parametrized by the variety (SX)∗ ' X (the hyperplane Lα
corresponding to a point α ∈ (SX)∗ is tangent to X along a quadric Qα ), and
the quadrics passing through a point x ∈ X are parametrized by an n2 -dimensional
quadric Qx ⊂ (SX)∗ .
2.13. Remark. Let X n be a Severi variety from 2.9. Arguing as in [47, pp. 234–
235] we obtain a resolution of singularities G ×P−M hQΛ i of the variety SX. This
n
resolution is a fiber bundle with fiber hQΛ i = P 2 +1 over the variety G/P−M ' X.
The projective fiber bundle PΛ ×PΛ ∩P−M hQΛ i of rank n2 + 1 over PΛ /PΛ ∩ P−M '
Q−M yields a resolution of singularities of the cone PΛ · hQΛ i = S(xΛ , X), and the
fiber bundle PΛ ×PΛ ∩P−M QΛ over the n2 -dimensional quadric Q−M whose fiber is
an n2 -dimensional quadric QΛ is birationally mapped onto X.
We turn to the study of varieties X n ⊂ PN for which SX = PN . Classification of
such varieties was given in Theorem 1.4 and Corollary 1.7. Apart from projective
spaces, the only varieties with small secant varieties (i. e. varieties X n for which
dim SX ≤ 2n or, which is the same, SX = T X) are Qn ⊂ Pn+1 , G(4, 1)6 ⊂ P9 ,
S 10 ⊂ P15 and P1 × Pn−1 ⊂ P2n−1 (n ≥ 3). From this and the structure of orbits
of the representation of G in V and the contragredient representation of G in V ∗
we derive the following result.
2.14. Proposition. Let X n ⊂ PN , PN = P(V ), N > n be a variety corresponding to a linear action of the group G on a vector space V for which SX = T X = PN .
Then there are the following possibilities:
1)
2)
3)
4)
{X n
{X n
{X n
{X n
⊂ PN } = {Qn ⊂ Pn+1 } (a nonsingular quadric);
⊂ PN } = {P1 × Pn−1 ⊂ P2n−1 } (a Segre variety);
⊂ PN } = {G(4, 1)6 ⊂ P9 } (the Grassmann variety);
⊂ PN } = {S 10 ⊂ P15 } (the spinor variety).
62
VARIETIES OF SMALL CODIMENSION CORRESPONDING TO GROUPS
In all these cases X is a self-dual variety, i. e. X ∗ ' X.
In [33] R. Hartshorne conjectured that for n > 23 N each nonsingular variety
X ⊂ PN is a complete intersection. In this connection it is natural to introduce
the following definition.
n
2.15. Definition. A nonsingular (not necessarily homogeneous) variety X n ⊂
PN is called a Hartshorne variety if n = 23 N and X is not a complete intersection.
2.16. Corollary. The Grassmann variety G(4, 1)6 ⊂ P9 and the spinor variety S 10 ⊂ P15 are the only Hartshorne varieties corresponding to orbits of linear
algebraic groups.
We describe the corresponding representations in more detail.
AH) G = SL5 , Λ = ϕ2 (or Λ = ϕ3 ), n = 6, N = 9, X = G(4, 1), IG [V ] = K,
N = V , and there are three orbits: {0}, the punctured cone of nonzero decomposable 2-vectors, and the set of indecomposable 2-vectors (of rank 4).
DH) G = Spin10 , Λ = ϕ5 (or Λ = ϕ4 ), n = 10, N = 15, X = S 10 , IG [V ] = K,
N = V , and there are three orbits: {0}, the punctured cone over X (the set of
nonzero ‘pure’ spinors), and the complement of (the closure of) this cone. The
variety S 10 parametrizes the four-dimensional linear subspaces from one family
on the eight-dimensional quadric. Furthermore, S 10 corresponds to the orbit of
highest weight vector of the spinor representation of the group B4 (Spin9 ) with
highest weight Λ = ϕ4 ; here IG [V ] = K[Q], where Q is a nondegenerate quadratic
form, and in N there are three orbits similar to the three orbits of the spinor
representation of D5 .
Using [3] and [8], we obtain the following result analogous to Proposition 2.10.
2.17. Proposition. For a Hartshorne variety X n corresponding to linear algebraic group the Hilbert polynomial has the form
¶nk
Y µt
H(t) =
+1
,
k
3n−2
4
k=1
where nk = 1 for k <
n+2
4
and k >
n
deg X =
n
2,
nk = 2 for
3n+2
4 ···n
n+2
n
4 ··· 2
n+2
4
≤k≤
n
2,
¡ n−2 ¢
! n!
¡
= n ¢ 4¡ 3n−2 ¢
!
!
2
4
(deg X 6 = 5, deg X 10 = 12). The Betti numbers of the Hartshorne variety X n
are given by the formula br = 0 for r ≡ 1 (mod 2), b0 = b2n = 1, b2k = 2 for
1 ≤ k ≤ n − 1, and the Euler characteristic e is equal to N + 1 = 32 n + 1. The
variety G(4, 1) is defined in P9 by five quadratic equations, and S 10 is defined in
P15 by ten quadratic equations.
2.18. While the n-dimensional Severi variety carries a family of
n
2 -dimensional
quadrics, the n-dimensional Hartshorne variety carries a natural family of (
n
−
2
2. HV –VARIETIES OF SMALL CODIMENSION
63
1)-dimensional linear subspaces. This family can be easily constructed using the
equality
n
codimPN ∗ X ∗ = codimPN X =
2
(to each point of X ∗ we associate the linear subspace along which the corresponding
hyperplane is tangent to X), but we shall show how to construct it using representation theory. In the notations of 2.11, the representation of H−M corresponding
to the restriction of Λ on the maximal torus of H−M has the form
1
◦—◦
◦
1
◦—◦—◦—◦
(SL3 × SL2 , ϕ2 ⊕ 0)
for G = SL6 ,
X = G(4, 1);
(SL5 , ϕ4 )
for G = Spin10 , X = S 10 .
It is clear that the orbit of xΛ under the action of H−M is a linear subspace PΛ
n
of dimension − 1 shifting which by means of the group G we obtain the desired
2
family of linear subspaces.
n
Apart from the above family of linear subspaces of dimension
− 1 the n2
dimensional homogeneous Hartshorne variety carries also a family of (n/2 + 1)dimensional quadrics. To construct this family we consider the orbit of xΛ with
respect to the action of the maximal subgroup Pe ⊂ G for which representation of
the corresponding semisimple group has the form
1
◦—◦—◦,
X = G(4, 1);
1
◦
◦—◦—◦
,
◦
X = S 10 .
It is clear that the orbit of xΛ under this action is an (n/2 + 1)-dimensional quadric
Qe . Shifting this quadric by means of the group G we obtain the desired family.
2.19. Proposition. On the Hartshorne variety X n ⊂ PN described in Corollary 2.16 there is a family of (n/2 − 1)-dimensional linear subspaces parametrized
by the variety X ∗ ' X. There is also a family of (n/2 + 1)-dimensional quadrics
on X (parametrized by P4 for n = 6 and by Q8 for n = 10) such that arbitrary two
points of X can be joined by a quadric from this family.
2.20. Remark. Let X n ⊂ PN be the Hartshorne variety from 2.16. Arguing as
in [47], we see that the variety PΛ ×PΛ ∩P−M PΛ is a projective fiber bundle of rank
n
n
−1 over P 2 −1 and the map PΛ ×PΛ ∩P−M PΛ → PΛ PΛ is birational for X = G(4, 1)
2
and is a bundle with fiber P1 for X = S 10 . Here PΛ PΛ = TX,xΛ ∩ X is a cone with
vertex xΛ whose base is P1 × P2 if X = G(4, 1) and G(4, 1) if X = S 10 (cf. also 3.1
and 3.6). The map G ×Pe hQe i → PN is birational, G/Pe ' P4 for X = G(4, 1) and
G/Pe ' Q8 for X = S 10 , PΛ · Qe = PΛ · Pe · xΛ = X, and the bundle PΛ ×PΛ ∩Pe Qe
n
n/2+1
with fiber Qe
over PΛ /PΛ ∩ Pe ' P 2 −1 is birationally mapped onto X.
64
VARIETIES OF SMALL CODIMENSION CORRESPONDING TO GROUPS
3. HV -varieties as birational images of projective spaces
As we already mentioned (cf. 1.1 and 1.3), from a general theorem of Rosenlicht
[72] it follows that HV -varieties are rational. However for HV -varieties one can
give an explicit geometric construction of birational isomorphism with projective
space.
3.1. Let X n = GxΛ ⊂ PN be an HV -variety, where PN = P(V ), G is a
semisimple group, G → Aut V is an irreducible representation, and xΛ ∈ PN is
the point corresponding to highest weight vector vΛ ∈ V (here Λ is the highest
weight). We denote by PΛ the stabilizer of the point xΛ (so that X ' G/PΛ ),
and let HΛ be the semisimple group corresponding to the parabolic subgroup PΛ .
The representation of HΛ in V obtained by restricting the representation of G is
reducible: it is clear that
HΛ · (KvΛ ),
HΛ · (gvΛ ) = gvΛ ,
where gvΛ is the tangent space to GvΛ at the point vΛ . Thus the HΛ -module V
can be represented in the form
V = KvΛ ⊕ t V ⊕ n V,
(3.1.1)
where KvΛ ⊕ t V = gvΛ is the tangent and n V is the ‘normal’ subspace to GvΛ ⊂ V
at the point vΛ ∈ GvΛ .
It is clear that the subset GvΛ ∩ (t V ⊕ n V ) is stable with respect to the action of
HΛ , and the stabilizer of an arbitrary point of GvΛ ∩ (t V ⊕ n ) contains a maximal
unipotent subgroup of HΛ . The corresponding projective variety H = X ∩ P(t V ⊕
n
V ) is the singular hyperplane section of X corresponding to the hyperplane P(t V ⊕
n
V ) ⊂ P(V ) which is tangent to X along the subvariety
Y = Sing H = X ∩ P(n V ).
(3.1.2)
3.2. Next we consider the rational projection
¯
πX = π ¯ : X n 99K Pn ,
Pn = P(KvΛ ⊕ t V )
X
with center in the (N − n − 1)-dimensional linear subspace P(n V ) ⊂ PN . Let
Pn−1 = P(t V ) ⊂ Pn = P(KvΛ ⊕ t V ),
and let A ⊂ Pn−1 ⊂ Pn be the projective variety corresponding to GvΛ ∩ t V . We
n
observe that, since X is defined in PN by quadratic equations (cf. [57]), TX,x
∩X is
Λ
n−1
a cone with vertex xΛ . It is clear that A ⊂ P
is the base of this cone. Under the
projection π, the hyperplanes passing through P(n V ) are mapped onto hyperplanes
in Pn . Furthermore,
¡
¢
π P(t V ⊕ n V ) = P(t V ) = Pn−1 ⊂ Pn ,
and
π(H) = A ⊂ Pn−1 .
The projection GvΛ → KvΛ ⊕ t V is a map of HΛ -spaces, and its fibers over points
lying in the same orbit are isomorphic to each
other. It is not hard to see that all
¯
ramification points of π lie in H , so that π ¯X\H is an isomorphism, π is a birational
isomorphism, and the variety X is rational.
3. HV –VARIETIES AS BIRATIONAL IMAGES OF PROJECTIVE SPACES
65
Summing up the discussion in 3.1 and 3.2, we obtain the following result.
3.3. Theorem. Let X n ⊂ PN be a projective HV -variety. Then there exist a
linear subspace M ⊂ PN , dim M = N − n − 1 and a hyperplane L ⊃ M such that
the projection π : X 99K Pn with
¯ center in M is a birational isomorphism. More
precisely, if H = L ∩ X, then π ¯
is an isomorphism and {π(H) ⊂ π(L)} = {A ⊂
X\H
Pn−1 = π(L) ⊂ Pn }.
3.4. For applications in Chapters IV and VI we need to know the structure
of representation R of the group HΛ in V for some specific representations of the
group G. We proceed with listing the results of our computations (in what follows
0 denotes the trivial representation, R(ψ) is the irreducible representation corresponding to weight ψ; in the case when HΛ is a simple group we denote by ϕi the
i-th fundamental weight in the notations of Bourbaki [9]; if the group HΛ is not
simple, then HΛ is a product of two simple groups, ϕi denotes the i-th fundamental
weight of one of them, and ϕ0i the i-th fundamental weight of the other one; the
first summand in the formulae below corresponds to the line KvΛ , the second to
the subspace t V and the third to the subspace n V ).
1) G = SLn+1 , Λ = 2ϕ1 , X = v2 (Pn ), HΛ = SLn , R = 0⊕R(ϕ1 )⊕R(2ϕ1 );
2) G = SLa+1 × SLb+1 , a, b ≥ 1, Λ = ϕ1 + ϕ01 , X = Pa × Pb , HΛ =
SLa × SLb , R = 0 ⊕ [R(ϕ1 ) + R(ϕ01 )] ⊕ R(ϕ1 + ϕ01 );
3) G = SLm+1 , m ≥ 3, Λ = ϕ2 , X = G(m, 1), HΛ = SL2 × SLm−1 , R =
0 ⊕ R(ϕ1 + ϕ01 ) ⊕ R(ϕ02 );
4) G = E6 , Λ = ϕ1 , X = E 16 , HΛ = Spin10 , R = 0 ⊕ R(ϕ5 ) ⊕ R(ϕ1 );
5) G = Spin10 , Λ = ϕ5 , X = S 10 , HΛ = SL5 , R = 0 ⊕ R(ϕ3 ) ⊕ R(ϕ1 ).
3.5. If the representation of HΛ in n V is irreducible, then its lowest weight vector
coincides with vM , where vM is the lowest weight vector of the representation of G
in V corresponding to the lowest weight M. Hence the variety Y corresponding to
the orbit of highest weight vector of the representation of HΛ in n V can be also
defined by the following formulae:
Y = P−Λ xM = w0 P−M w0 xM = w0 P−M xΛ ,
where P−Λ is the stationary subgroup of the lowest weight vector of the contragredient representation of G in V ∗ . This allows to compute Y in cases 1)–5) from 3.4:
1) Y = v2 (Pn−1 ), P(n V ) = hY i = P
2) Y = P
a−1
×P
b−1
n(n−1)
−1
2
n
, P( V ) = hY i = P
n
3) Y = G(m − 2, 1), P( V ) = hY i = P
8
n
ab−1
;
;
m(m−3)
2
;
9
4) Y = Q , P( V ) = hY i = P ;
5) Y = P−M PΛ = P4 .
3.6. From 1.3 it follows that the representation of HΛ in t V is irreducible, and
it is clear that A is an HΛ -variety. Using the explicit form of representation of HΛ
in t V computed in 3.4 it is easy to describe A in cases 1)–5):
66
VARIETIES OF SMALL CODIMENSION CORRESPONDING TO GROUPS
1) A = ∅;
2) A = Pa−1
`
Pb−1 ;
3) A = P1 × Pm−2 ;
4) A = S 10 ;
5) A = G(4, 1).
¯
In case 1) π is an isomorphism, and in cases 2)–5) π ¯X\H is an isomorphism and
¯
π ¯H is a rational bundle with fiber:
2) Pb over Pa−1 , Pa over Pb−1 ;
3) Pm−2 ;
4) P5 ;
5) P3 .
3.7. Using tables in the end of [94] it is easy to verify that in cases 1)–5) from
3.4 we have the following formulae (S 2 denotes the second symmetric power):
¡
¢
1) S 2 0 ⊕ R(ϕ1 ) ' 0 ⊕ R(ϕ1 ) ⊕ R(2ϕ1 );
¡
¢
2) S 2 0 ⊕ [R(ϕ1 ) ⊕ R(ϕ01 )] ' 0 ⊕ [R(ϕ1 ) ⊕ R(ϕ01 )] ⊕ R(ϕ1 + ϕ01 ) ⊕ [R(2ϕ1 ) ⊕
R(2ϕ01 )];
¢
¡
3) S 2 0 ⊕ R(ϕ1 + ϕ01 ) ' 0 ⊕ R(ϕ1 + ϕ01 ) ⊕ R(ϕ02 ) ⊕ R(2ϕ1 + ϕ01 );
¡
¢
4) S 2 0 ⊕ R(ϕ5 ) ' 0 ⊕ R(ϕ5 ) ⊕ R(ϕ1 ) ⊕ R(2ϕ5 );
¡
¢
5) S 2 0 ⊕ R(ϕ3 ) ' 0 ⊕ R(ϕ3 ) ⊕ R(ϕ1 ) ⊕ R(2ϕ3 ).
It is easy to see that in these formulae the first term corresponds to the summand
KvΛ , the second term to the summand t V , and the third one to the summand n V
in decomposition (3.1.1). From this and 3.2 it follows that the linear system of
quadrics in TX,xΛ containing the subvariety A ⊂ Pn−1 ⊂ TX,xΛ defines a rational
map
σ : Pn 99K P(KvΛ ⊕ t V ⊕ n V ),
which is inverse to π. From formulae 1)–5) it also follows that all the above varieties
are rational projections of the Veronese varieties v2 (Pn ), viz.
1) X = v2 (Pn );
2) the Segre variety Pa × Pb is obtained from v2 (Pa+b ) by projecting it from
hv2 (Pa−1 ), v2 (Pb−1 )i, Pa−1 ∩ Pb−1 = ∅;
3) the Grassmann variety G(m, 1) is obtained from v2 (P2m−2 ) by projecting
it from hv2 (P1 ) × v2 (Pm−2 )i;
4) E 16 is obtained from v2 (P16 ) by projecting it from hv2 (S 10 )i = P125 ;
5) S 10 is obtained from v2 (P10 ) by projecting it from hv2 (G(4, 1)i = P49 .
Summing up the discussion in 3.4–3.7, we obtain the following result.
3. HV –VARIETIES AS BIRATIONAL IMAGES OF PROJECTIVE SPACES
67
n(n+3)
3.8. Theorem. 1) Let X = v2 (Pn ) ⊂ P 2 . The projection π : X → Pn
with center in hv2 (Pn−1 )i is an isomorphism inverse to the Veronese map
σ = v2 : Pn → X.
2) Let X = Pa × Pb ⊂ Pab+a+b . The projection π : X 99K Pa+b with center in
hPa−1 × Pb−1 i is ¯a birational isomorphism; moreover, if H = Pa × Pb−1 ∩
Pa−1 × Pb , then π ¯X\H is an isomorphism. The inverse map σ : Pa+b 99K X
`
is defined by the linear system of quadrics passing through Pa−1 Pb−1 ⊂
Pa+b−1 ⊂ Pa+b . The variety X is obtained from the Veronese variety
v2 (Pa+b ) by projecting it from hv2 (Pa−1 ), v2 (Pb−1 )i.
m2 +m−2
3) Let X = G(m, 1) ⊂ P 2
. The projection π : X 99K P2(m−1) with
center in hG(m − 2, 1)i is a birational isomorphism; moreover, if¯ H is the
Schubert divisor corresponding to a subspace Pm−2 ⊂ Pm , then π ¯X\H is an
isomorphism. The inverse map σ : P2(m−1) 99K X is defined by the linear
system of quadrics containing the Segre variety P1 × Pm−2 ⊂ P2m−3 ⊂
P2(m−1) . The variety X is obtained from the Veronese variety v2 (P2(m−1) )
by projecting it from hv2 (P1 ) × v2 (Pm−2 )i.
4) Let X = E 16 ⊂ P26 . The projection π : X 99K P16 with center in hQ8 i = P9 ,
where Q8 ⊂ P16 is a nonsingular quadric, is a birational isomorphism;
moreover,
if H is the hyperplane section of X such that Sing H = Q8 , then
¯
π ¯X\H is an isomorphism. The inverse map σ : P16 99K X is defined by the
linear system of quadrics containing the spinor variety S 10 ⊂ P15 ⊂ P16
parametrizing four-dimensional linear subspaces from one family on the
eight-dimensional quadric. The variety X is obtained from the Veronese
variety v2 (P16 ) ⊂ P152 by projecting it from hv2 (S 10 )i = P125 .
5) Let X = S 10 ⊂ P15 . The projection π : X 99K P10 with center in a linear subspace P4 ⊂ X is a birational
¯ isomorphism; moreover, if H is a
singular hyperplane section, then π ¯X\H is an isomorphism. The inverse
map σ : P10 99K X is defined by the linear system of quadrics containing the
Grassmann variety G(4, 1) ⊂ P9 ⊂ P10 . The variety X is ¡obtained¢ from the
Veronese variety v2 (P10 ) ⊂ P65 by projecting it from hv2 G(4, 1) i = P49 .
3.9. Remark. In Theorem 3.8 we described projections of those HV -varieties
which
will be discussed in Chapters IV and VI and analyzed the structure of maps
¯
π ¯H and σ = π −1 . Some other HV -varieties for which H may contain more than two
orbits of the group HΛ and σ has a more complex structure also present geometric
interest. A direct application of Theorem 3.3 shows that for all d > 1 projection of
the Veronese variety vd (Pn ) with center in a subspace hvd (Pn−1 )i is an isomorphism
inverse to vd ; the Grassmann variety G(m, k) can be birationally projected onto
P(k+1)(m−k) , and the fundamental subset A = π(H) of the map σ coincides with
the Segre variety
Pk × Pm−k−1 ⊂ P(k+1)(m−k)−1 ⊂ P(k+1)(m−k) ;
the spinor variety Sk parametrizing the k-dimensional linear subspaces from one
family on a nonsingular 2k-dimensional quadric Q2k ⊂ P2k+1 (Sk corresponds to
68
VARIETIES OF SMALL CODIMENSION CORRESPONDING TO GROUPS
the orbit of highest weight vector of the spinor representation of the group Dk+1 =
k(k+1)
Spin2k+2 ) can be birationally projected onto P 2 , and the fundamental subset
k(k+1)
A = π(H) of the map σ coincides with the Grassmann variety G(k, 1) ⊂ P 2 −1 ⊂
k(k+1)
P 2 , etc. Nonsingular hyperplane sections of Severi varieties (examples A0 ), A2 ),
C), and F) from § 2) can also be interpreted in this way (for such a variety A is a
hyperplane section of the variety A for the corresponding Severi variety).
3.10. Remark. In the case of Segre variety P2 ×P2 and Grassmannians existence
of a birational projection π : X 99K Pn was proved by different methods in the
classical papers [79] and [81].
CHAPTER IV
SEVERI VARIETIES
Typeset by AMS-TEX
69
70
IV. SEVERI VARIETIES
1. Reduction to nonsingular case
1.1. The goal of this chapter is to give classification of extremal varieties with
small secant varieties, i.e. varieties for which the inequality in Theorem 2.8 of Chapter II turns into equality. In other words, we classify nondegenerate varieties
X n ⊂ PN ,
n=
2N + b
− 1,
3
b = dim (Sing X)
(1.1.1)
which can be J-isomorphically projected to PN −1 . By Proposition 1.5 of Chapter II,
the last condition holds if and only if SX 6= PN (in view of Remark 2.10 in Chapter II, for n > 1 this condition can be replaced by the condition T 0 X 6= PN which,
according to Proposition 1.5 of Chapter II, ensures the existence of a J-unramified
projection of X to PN −1 . From Theorem 2.8 of Chapter II it follows that under
these assumptions
3n − b + 1
dim SX = N − 1 =
.
(1.1.2)
2
Moreover, from Theorem 2.3 of Chapter V and Theorems 1.4 and 4.7 of the present
chapter it follows that if
0
X 0 ⊂ PN ,
0
SX 0 6= PN ,
dim (Sing X 0 ) = b,
dim X 0 = n,
3n − b + 1
dim SX 0 =
2
(1.1.3)
(a priori one can only claim
is a nondegenerate variety, then N 0 = N = 3(n+1)−b
2
that any variety X 0 satisfying (1.1.3) can be J-isomorphically projected onto a
variety X satisfying (1.1.1)).
Throughout this chapter we consider varieties defined over an algebraically closed
field K, char K = 0. We recall the following definition (cf. Definition 2.3 in Chapter III).
1.2. Definition. A nondegenerate nonsingular variety X n ⊂ PN , n = 23 (N − 2)
is called Severi variety if X can be isomorphically projected to PN −1 .
In view of Proposition 1.5 b), d) and Corollary 1.7 from Chapter II, a nonsingular
nondegenerate variety X n ⊂ PN , n = 32 (N − 2) is a Severi variety if any of the
following equivalent conditions holds:
a) SX 6= PN ;
b) there exists an unramified projection of X to a projective space of smaller
dimension;
c) T X 6= PN .
1.3. Remark. Severi varieties are named after Francesco Severi who gave their
classification in the case n = 2 (cf. [82] and also [62; 15]). More historical details are
given in Remark 4.11. We recall that in Chapter III (cf. Theorem 2.9) we gave four
examples of Severi varieties in dimensions 2, 4, 8, and 16, viz. the Veronese surface
v2 (P2 ) ⊂ P5 , the Segre variety P2 × P2 ⊂ P8 , the Grassmann variety G(5, 1)8 ⊂ P14 ,
and the variety E 16 ⊂ P26 corresponding to the orbit of highest weight vector of
the simplest nontrivial representation of the group E6 . In Theorem 4.7 we show
that these are the only Severi varieties.
1. REDUCTION TO NONSINGULAR CASE
71
1.4. Theorem. Let X n ⊂ PN , SX 6= PN be a nondegenerate variety satisfying
condition (1.1.1) (so that X has maximal possible dimension for given N and b).
Then X is a projective cone with vertex Pb = Sing X whose base is a Severi variety
X0n−b−1 ⊂ PN −b−1 . Conversely, let X0n0 ⊂ PN0 be a Severi variety, let b ≥ 0 be an
integer, and let X n ⊂ PN , n = n0 + b + 1, N = N0 + b + 1 be the projective cone
over X0 with vertex Pb . Then SX 6= PN , Sing X = Pb , and n = 2N3+b − 1.
Proof. We already observed that SX is a hypersurface (cf. (1.1.2)). Let z be a
general point of SX, and let Yz = p1 (ϕ−1 (z)) (cf. Chapter II, (2.8.1)). Since for
our variety X the inequality in Theorem 2.8 of Chapter II turns into equality, from
the proof of this theorem it immediately follows that dim (Yz ∩ Sing X) = b, i.e.
there exists an irreducible component Ξ of the variety Sing X such that
\
Yz
(1.4.1)
dim Ξ = b,
Ξ⊂
z∈SX
(cf. Remark 2.13 in Chapter II). From Proposition 1.9 of Chapter II it follows that
Yz ⊂ TSX,z .
(1.4.2)
Combining (1.4.1) and (1.4.2), we see that
\
Ξ⊂
TSX,z .
(1.4.3)
z∈SX
Since char K = 0, from (1.4.1) and (1.4.3) it follows that SX is a cone with vertex
Ξ = Pb ⊂ X ⊂ SX
(1.4.4)
(to verify this without computations it suffices to refer to C. Segre’s reflexivity
theorem (cf. e.g. [49; 50])).
Let x ∈ Ξ, let y, y 0 be a general pair of points of X, and let z be a general point
of the chord hy, y 0 i. Since SX is a cone with vertex x, TSX,z 0 = TSX,z for all points
z 0 ∈ hx, zi r x. Therefore from Proposition 1.9 a) of Chapter II it follows that the
hyperplane TSX,z is tangent to X at all points of Yz ∩ Sm X, where
Yz = p1 (ϕ−1 (hx, zi r x)) =
[
Yz 0 .
z 0 ∈hx,zirx
Applying the arguments used in the proof of Theorem 2.8 of Chapter II to the
subvariety Yz (or applying Theorem 1.7 of Chapter I to the subvariety π(Yz ) ⊂
π(X) ⊂ Pn−1 and the hyperplane π(TSX,z ) ⊂ PN −1 , where π is the projection with
center in a general point of TSX,z \ SX), we see that
dim Yz = dim Yz =
n+b+1
= 2n − N + 2.
2
Thus Yz consists of components of Yz , and we may assume that
Yz 0 ⊃ Yz
(1.4.5)
72
IV. SEVERI VARIETIES
for z 0 ∈ hx, zi. From (1.4.5) it follows that
¡
¢
C = p2 (p1 × ϕ)−1 (y 0 × hx, zi) ⊂ X ∩ Π
is a one-dimensional subvariety of the plane Π = hx, y 0 , zi. Without loss of generality
we may assume that
hx, zi ∩ C 1 = hx, zi ∩ X = x,
where C 1 is a one-dimensional component of C passing through y. In fact, otherwise
SX = S(x, X) and therefore
dim Yz = 2n + 1 − dim SX = 2n + 1 − dim S(x, X) = n,
so that Yz = X and from (1.4.2) it follows that X ⊂ TSX,z contrary to the assumption that X is nondegenerate. From (1.4.6) it follows that a general line in Π
passing through x intersects C 1 only at x, and therefore C 1 = hx, yi (cf. e.g. [64,
5.11]).
Thus for a general and therefore for each point y ∈ X we have hx, yi ⊂ X, i.e.
X is a cone with vertex x. Since x is an arbitrary point of Ξ, from this it follows
that X is a cone with vertex Ξ = P b, and since X is irreducible, b < n − 1.
Let M N0 ⊂ PN be a general linear subspace, M ∩ Sing X = ∅, N0 = N − b − 1,
X0 = X ∩ M , n0 = dim X0 = n − b − 1 > 0. By Bertini’s theorem (cf. [28,
Vol. I, Chapter I, § 1; 34, Chapter II, 8.18]), X0 is irreducible and nonsingular.
Furthermore,
SX0 ⊂ SX ∩ M 6= M
and
µ
n0 = n − b − 1 =
¶
2N + b
2(N − b − 1) − 4
2
−1 −b−1=
= (N0 − 2),
3
3
3
i.e. X0 is a Severi variety.
The converse is an immediate consequence of the fact that, as it is easy to see,
SX is the cone over SX0 with vertex Pb = Sing X, and since n0 = 23 (N0 − 2),
n = n0 + b + 1 =
¤
2(N0 + b + 1) + b − 3
2N + b
=
− 1.
3
3
2. QUADRICS ON SEVERI VARIETIES
73
2. Quadrics on Severi varieties
2.1. Proposition. Let X n ⊂ PN , n = 23 (N − 2) be a Severi variety, and let
¯
z be a general point of SX. Set L = TSX,z , P L = {u ∈ SX ¯ TSX,u = L}. Then
X ∩P L = Yz = p1 (ϕ−1 (z)) is a nonsingular n2 -dimensional quadric in the projective
n
space P L = P 2 +1 .
Proof. According to C. Segre’s reflexivity theorem (cf. [49; 50]), P L is a linear
subspace of PN .
Let
¯
XL = {x ∈ X ¯ TX,x ⊂ L}.
Then XL is a closed subvariety of X, T (XL , X) ⊂ L, and since X is nondegenerate,
S(XL , X) 6⊂ L. Hence Theorem 1.4 of Chapter I shows that
dim SX =
3n
+ 1 ≥ dim S(XL , X) = dim XL + n + 1,
2
so that
n
.
(2.1.1)
2
On the other hand, for a general point u ∈ P L from Proposition 1.9 of Chapter II
it follows that
dim XL ≥
def
Yu = p1 (ϕ−1 (u)),
T (Yu , X) ⊂ TSX,u = L,
i.e.
Yu ⊂ XL .
(2.1.2)
Since X is a Severi variety, we have
µ
dim Yz = dim Yu = 2n + 1 − dim SX = 2n + 1 −
3n
+1
2
¶
=
n
.
2
(2.1.3)
Combining (2.1.1), (2.1.2), and (2.1.3), we see that for a general point u ∈ P L we
have Yu = Yz . Therefore
SYz = P L
(2.1.4)
and by (2.1.3)
2 dim Yz + 1 − dim P L = dim Yz ,
dim P L = dim Yz + 1 =
n
n
+ 1,
2
n
so that P L = P 2 +1 and Yz is a hypersurface in P 2 +1 . Hence for a general point
u ∈ P L there is an inclusion Yu ⊃ X ∩ P L , and therefore
X ∩ P L = Yz .
(2.1.5)
From (2.1.4) it follows that deg Yz ≥ 2. From the trisecant lemma (cf. [39, 2.5]
and also [34, Chapter IV, § 3] and [64, § 7 B]) it follows that for a general pair of
points x, y ∈ X
hx, yi ∩ X = {x, y}.
(2.1.6)
74
IV. SEVERI VARIETIES
Since z is a general point of SX, from (2.1.6) it follows that deg Yz = 2.
It remains to verify that the quadric Yz is nonsingular. Suppose that this is
not so. Then Yz is a quadratic cone with vertex at a point y ∈ Yz . Therefore
SX = S(Yz , X) is a cone with vertex y, and for a general point u ∈ SX we have
y ∈ PM , M = TSX,u . In view of (2.1.5), from this it follows that
\
y∈
Yu .
(2.1.7)
u∈SX
From (2.1.2) and (2.1.7) it follows that TX,y ⊂
vertex TX,y and
TX,y ⊂
\
T
u
TSX,u , i.e. SX is a cone with
PM
M
which is impossible since
dim PM =
n
+ 1 ≤ n = dim TX,y .
2
This contradiction shows that Yz is a nonsingular quadric.
¤
2.2. Lemma. Let X n ⊂ PN , n = 23 (N − 2) be a Severi variety, and let z
be a general point of SX. Then, in the notations of Section 1 of Chapter I, the
morphism ϕz = ϕYz : SYz ,X → S(Yz , X) = SX, Yz = p1 (ϕ−1 (z)) is birational.
Proof. From Proposition 1.9 a) of Chapter II it follows that
N −1
T (Yz , X) ⊂ TSX,z
,
and since
hS(Yz , X)i = hXi = PN ,
Theorem 1.4 of Chapter I shows that
dim S(Yz , X) = dim SYz ,X = dim SX = N − 1 =
3n
+ 1.
2
Hence S(Yz , X) = SX and the morphism ϕz is generically finite, i.e. for a general
point u ∈ SX we have card (Yz ∩ Yu ) < ∞. To prove Lemma 2.2 it suffices to
verify that for a general point u ∈ SX the quadrics Yz and Yu (cf. Proposition 2.1)
transversely intersect at a unique point (Yz ∩ Yu 6= ∅ since u ∈ S(Yz , X) = SX).
Let Pz (resp. Pu ) be the ( n2 + 1)-dimensional linear subspace spanned by the
quadric Yz (resp. Yu ). Suppose that Yz ∩ Yu 3 x, y, and let l = hx, yi (in the case
when y = x, i.e. Yz and Yu are tangent at x, l is their common tangent line). Then
l ⊂ Pz ∩ Pu ⊂ Sing (SX)
since by Proposition 2.1 the tangent space to SX at an arbitrary point of Pz ∩ Pu
contains both TSX,z and TSX,u 6= TSX,z . Varying u ∈ SX, we see that a general
point y ∈ Yz lies on a line ly ⊂ Pz ∩ (Sing X \ X), from which it follows that
z ∈ Pz ⊂ Sing SX contrary to the choice of z. This contradiction completes the
proof of Lemma 2.2. ¤
2. QUADRICS ON SEVERI VARIETIES
75
2.3. Lemma. Let X n ⊂ PN , n = 32 (N − 2) be a Severi variety. Then the
quasiprojective variety SX \ X has nonsingular normalization.
Proof. Let v ∈ Sing (SX \ X), and let Yv0 be an irreducible component of Yv =
p1 (ϕ−1 (v)). Then either
S(Yv0 , X) 6= SX
(2.3.1)
and for a general point z ∈ SX
or
Yz ∩ Yv0 = ∅
(2.3.2)
S(Yv0 , X) = SX.
(2.3.3)
We claim that in the case (2.3.3)
dim Yv0 =
n
.
2
(2.3.4)
Suppose that this is not so and
dim Yv0 > 2n + 1 − dim SX =
n
.
2
(2.3.5)
In view of (2.3.3) and Theorem 1.4 of Chapter I, from (2.3.5) it follows that
T (Yv0 , X) = SX.
(2.3.6)
Remark 1.10 in Chapter II shows that if v ∈ hy, xi, y ∈ Yv0 , x ∈ X, x 6= y, then
0
0
Tv,S(x,X)
⊃ TX,y , where Tv,S(x,X)
is the tangent cone to S(x, X) at the point v (we
use the notations of § 1 of Chapter I). Therefore
0
0
Tv,SX
⊃ Tv,S(x,X)
⊃ TX,y .
0
On the other hand, if v ∈ TX,y , then it is clear that Tv,SX
⊃ TX,y . Thus
0
T (Yv0 , X) ⊂ Tv,SX
.
(2.3.7)
0
Combining (2.3.6) and (2.3.7) we see that SX ⊂ Tv,SX
, i.e. SX is a cone with
vertex v. Hence, in the notations of Lemma 2.2,
\
v∈
Pz .
(2.3.8)
z∈X
But in the proof of Lemma 2.2 we verified that for a general pair of points z, u ∈ SX
the variety Pz ∩Pu = Yz ∩Yu reduces to a unique point lying in X, which contradicts
(2.3.8). This contradiction proves (2.3.4).
From (2.3.2) and (2.3.4) it follows that for a general point z ∈ SX we have
card (Yz ∩ Yu ) < ∞, i.e. in the notations of Lemma 2.2 ϕ−1
z (v) is a finite set. Hence
Lemma 2.2 and Proposition 2.1 show that in a neighborhood of v the normalization
of SX = S(Yz , X) is isomorphic to SYz ,X and therefore is nonsingular. Lemma 2.3
is proved. ¤
76
IV. SEVERI VARIETIES
2.4. Theorem. Let X n ⊂ PN , n = 32 (N − 2) be a Severi variety. Then
a) SX is a normal hypersurface and Sing SX = X;
b) Each point z ∈ SX \ X is general in the sense of Proposition 2.1, i.e. Yz is
n
a ¯nonsingular quadric in the projective space Pz = P 2 +1 ;
c) ϕ ¯ϕ−1 (SX\X) is a smooth morphism, and the tangent spaces at arbitrary
points x, y ∈ X such that the line hx, yi does not lie in X span a hyperplane,
i.e. dim hTX,x , TX,y i = N − 1 = 3n
2 + 1;
d) An arbitrary secant of the variety X either lies on X or intersects X at
exactly two points (which may coincide with each other);
e) SX is a cubic and multx SX = 2 for each point x ∈ X = Sing SX (here
multx SX denotes the multiplicity of x on SX);
f) For an arbitrary point z ∈ SX \ X the projection with center in Pz defines
a birational map πz : X 99K P n which is an isomorphism outside X ∩ TSX,z .
Proof. a) Let
ϕ = ν ◦ϕ̃,
g
ϕ̃ : SX → SX,
g → SX
ν : SX
be the Stein factorization of the morphism ϕ : SX → SX. From Proposition 2.1 it
g → SX is the normalization morphism. Lemma 2.3 shows that
follows that ν : SX
g ⊂ ν −1 (X).
Sing SX
(2.4.1)
Suppose that Sing SX 6= X, let v be a general point of Sing (SX \ X), and let
ṽ ∈ ν −1 (v). In view of (2.4.1) and the Serre normality criterion (cf. e.g. [30,
Chapter IV2 , (5.8.6)]),
¡
¢
3n
dim Sing (SX \ X) = dim SX − 1 =
.
2
Hence dim ϕ̃−1 (v) = n2 and ϕ̃∗ ([ṽ]) ≡ [ϕ−1 (z)], where z is a general point of SX,
brackets denote the cycle corresponding to subvariety, and ≡ denotes algebraic
equivalence. Therefore
¡
¢
¡
¢
(p1 )∗ ϕ̃∗ ([v]) = (p1 )∗ [ϕ−1 (z)] = [Yz ].
(2.4.2)
By Proposition 2.1 and Lemma 2.2, Yz is a nonsingular quadric and
¡
Yz2
¢
X
= 1.
(2.4.3)
Hence all components of Yṽ = p1 (ϕ̃−1 (ṽ)) are also n2 -dimensional quadrics. From
(2.4.2) and (2.4.3) it follows that for a general point v ∈ Sing X \X the fiber ϕ−1 (v)
is connected, i.e. the morphism ν is one-to-one. Since for x ∈ X the fiber ϕ−1 (x)
is connected and contains a reduced component (which is isomorphic to X), from
this it follows that ν is an isomorphism, i.e. SX is a normal hypersurface. From
Lemma 2.3 it follows that Sing SX ⊂ X. On the other hand, Lemma 1.8 a) from
Chapter II shows that X ⊂ Sing SX. Thus X = Sing SX. Assertion a) is proved.
2. QUADRICS ON SEVERI VARIETIES
77
b) Let z ∈ SX \ X be an arbitrary point. Then from a) and Proposition 1.9 a)
in Chapter II it follows that
T (Yz , X) ⊂ TSX,z 6= PN .
Since S(Yz , X) ⊃ X,
S(Yz , X) 6= T (Yz , X),
and from Theorem 1.4 of Chapter I it follows that
dim S(Yz , X) = dim Yz + n + 1,
S(Yz , X) = SX,
dim Yz =
n
.
2
(2.4.4)
We already proved in a) that under these conditions Yz is a quadric. Moreover, in
the proof of Proposition 2.1 it is shown that from (2.4.4) it easily follows that the
quadric Yz is nonsingular. Assertion b) is proved.
c) The first claim follows from b), and the second claim is a consequence of the
first one and the proof of Proposition 1.9 b) in Chapter II.
d) immediately follows from b).
¡
¢
e) Let x be an arbitrary point of X, and let Qx = p1 ψ −1 (x) (we use the
notations of § 1 of Chapter I). We observe that S(Qx , X) 6= SX. In fact,
T otherwise
from Proposition 1.9 a) of Chapter II it would follow that x ∈
TSX,z , i.e.
z∈SX
SX is a cone with vertex x, which is impossible in view of b) and the proof of
Proposition 2.1.
n
2 +2
Let z ∈ SX \ S(Qx , X), and let Pz,x
= hPz , xi. Then
Pz,x ∩ X = Yz ∪ x,
n
+2
2
and the intersection of subvarieties Pz,x
3n
+2
2
space P
is transverse. In fact, if
x 0 ∈ Pz,x ∩ X,
(2.4.5)
and X n at the point x of the projective
x0 ∈
/ Yz ,
u ∈ hx, x 0 i ∩ Pz ,
then in view of the choice of z
x∈
/ T (Yz , X)
(2.4.6)
and from d) it follows that u ∈
/ X, so that b) yields Yu = Yz which is impossible by
(2.4.6) since it is clear that Yu 3 x.
Now let u ∈ Pz,x ∩ SX be an arbitrary point. As we already observed, Yu ∩ Yz 6=
∅, and either
u ∈ TX,y ∩ Pz,x ,
y ∈ Yz
(2.4.7)
or
But
u ∈ hy, x 0 i,
y ∈ Yz ,
x 0 ∈ X ∩ Pz,x .
n
(TX,y ∩ Pz,x ) ⊂ (T (Yz , X) ∩ Pz,x ) = Pz2
+1
,
(2.4.8)
(2.4.9)
78
IV. SEVERI VARIETIES
and from (2.4.5), (2.4.7), (2.4.8), and (2.4.9) it follows that
Pz,x ∩ SX = Pz ∪ Sx Yz .
(2.4.10)
Assertion e) immediately follows from (2.4.10).
f) immediately follows from the proof of assertion e).
¤
2.5. Remark. From Theorem 2.4 b) it follows that
S(SX)∗ ⊂ X ∗ .
(2.5.1)
Counting dimensions, we see that each hyperplane in PN which is tangent to X at
a point x ∈ X lies in the pencil generated by two hyperplanes which are tangent to
SX at some points of S(x, X) (cf. Proposition 3.1 for more details). Therefore
¡
¢
S (SX)∗ = X ∗ .
(2.5.2)
Applying Theorem 2.4 (and specifically assertion e) of this theorem), it is not hard
to show that the variety X = Sing SX is an intersection of quadrics and the linear
system of quadrics passing through X defines a birational map
κX : PN 99K PN ,
(2.5.3)
under which SX is transformed to X and vice versa (if F (x0 : · · · : xN ) = 0 is
the equation
SX, then κX is defined by the formula
µ of the cubic hypersurface
¶
∂F
∂F
κX (x) =
(x) : · · · :
(x) ). Since by definition κX (SX) = X ∗ , from this
∂x0
∂xN
it follows that
X ∗ ' SX,
(SX)∗ ' X.
(2.5.4)
3. DIMENSION OF SEVERI VARIETIES
79
3. Dimension of Severi varieties
3.1. Proposition. Let X n ⊂ PN , N = 23 (N − 2)¯ be a Severi variety, let
x ∈ X be an arbitrary point, and let Y x = {(TSX,z )∗ ¯ z ∈ S(x, X) \ X} (here
(TSX,z )∗ ∈ PN ∗ is the point corresponding to the hyperplane TSX,z ). Then Y x is a
nonsingular quadric in the ( n2 + 1)-dimensional projective space (TX,x )∗ ⊂ PN ∗ .
Proof. It is clear that Y x is an irreducible variety. Since dim S(x, X) = n + 1
and for z ∈ S(x, X) \ X the hyperplane TSX,z is tangent to SX along the ( n2 + 1)dimensional linear subspace
Pz = SYz ⊂ S(x, X),
we conclude that
n
n
+ 1) = .
2
2
From Theorem 2.4 it follows that Y x is a (closed) irreducible hypersurface in
(TX,x )∗ . Since the variety (SX)∗ ⊂ PN ∗ is not a hypersurface (cf. e.g. (2.5.1)),
from the trisecant lemma (cf. [39, 2.5] and also [34, Chapter IV, § 3] and [64, §7 B])
it follows that deg Y x = 2.
It remains to show that the quadric Y x is nonsingular. In fact, otherwise the
quadric Y x would be a cone, so that there would exist a point z ∈ S(x, X) \ X such
that
dim (Yz ∩ Yu ) > 0 ∀u ∈ S(x, X) \ X,
dim Y x = (n + 1) − (
which contradicts Theorem 2.4 f).
¤
3.2. Proposition. Let X n ⊂ PN , n = 23 (N − 2) be a Severi variety, and let
z1 , z2 ∈ SX \ X. Then either z1 ∈ Pz2 , z2 ∈ Pz1 and Yz1 = Yz2 , Pz1 = Pz2 or
Yz1 ∩ Yz2 is a linear subspace (we use the notations from the proof of Lemma 2.2).
Proof. As we have already observed,
S(Yzi , X) = SX,
i = 1, 2,
and therefore Yz1 ∩ Yz2 6= ∅. Suppose that Yz1 ∩ Yz2 is not a linear subspace. Then
from Theorem 2.4 it follows that
S(Yz1 ∩ Yz2 ) = Pz1 ∩ Pz2 6⊂ X.
But for an arbitrary point z ∈ S(Yz1 ∩ Yz2 ) \ X from Theorem 2.4 b) it follows that
Yz1 = Yz = Yz2 .
¤
Let X n ⊂ PN , N = 23 (n − 2) be a Severi variety, let x be an arbitrary point of
X, and let
Y1 = Yz1 ,
Y2 = Yz2 ,
z1 , z2 ∈ S(x, X) \ X
be two quadrics for which (Y1 · Y2 ) = x (cf. Lemma 2.2). Put
i = 1, 2.
Ci = Yi ∩ TX,x = Yi ∩ TYi ,x ,
n
Then Ci is an ( 2 −1)-dimensional cone with vertex x in the n2 -dimensional projective
space TYi ,x whose base is a nonsingular ( n2 − 2)-dimensional quadric (i = 1, 2). It
is clear that for n > 2, S(C1 , C2 ) 6⊂ X (here, as in § 1 of Chapter I, S(C1 , C2 ) is the
join of cones C1 and C2 ).
80
IV. SEVERI VARIETIES
3.3. Proposition. a) dim S(C1 , C2 ) = n − 2;
b) Let n > 2, z ∈ S(C1 , C£2 ) ¤\ X, Yz 6= Yi (i = 1, 2). Then Yz ∩ Yi is a linear
subspace of dimension n4 ;
c) For n > 2 we have n ≡ 0 (mod 4);
d) For n > 4 we have n ≡ 0 (mod 8).
Proof. a) Let Qi be the base of the cone Ci (i.e. Qi is the intersection of Ci with
a general hyperplane in TYi ,x , i = 1, 2). Then
S(C1 , C2 ) = S(x, S(Q1 , Q2 ))
and
hQ1 i ∩ hQ2 i = ∅.
Hence
dim S(C1 , C2 ) = dim S(Q1 , Q2 ) + 1 = 2(
n
− 2) + 2 = n − 2.
2
Assertion a) is proved.
b), c). Let z be a general point of S(C1 , C2 ) \ X. By Proposition 3.2,
Yz ∩ Yi = P αi ,
i = 1, 2.
Since C1 and C2 are cones with vertex x, αi > 0 (i = 1, 2) and
P αi 3 x, i = 1, 2
P α1 ∩ P α2 = C1 ∩ C2 = Y1 ∩ Y2 = x,
z ∈ S(P α1 , P α2 ).
Furthermore, if z 0 ∈ S(C1 , C2 ) \ X, then Yz 0 = Yz if and only if
z 0 ∈ S(P α1 , P α2 ) = P α1 +α2 \ X.
(3.3.1)
¡
¢
By a) and (3.3.1), varying z ∈ S(C1 , C2 ) \ X we obtain an (n − 2) − (α1 + α2 ) dimensional family of quadrics passing through x and intersecting Y1 and Y2 along
linear subspaces of positive dimension.
We already know (cf. Proposition 3.1) that there is an n2 -dimensional family of
quadrics Yu passing through x and parametrized by a quadric Y x . Furthermore,
the ( n2 − 1)-dimensional subfamily of quadrics Yu intersecting Y1 along a positivedimensional linear subspace is parametrized by the subcone with vertex (TSX,z1 )∗
in Y x . From this it follows that the dimension of the family of quadrics Yu passing
through x and intersecting Y1 and Y2 along positive-dimensional linear subspaces
is equal to n2 − 2 (the base Y12 of this family is the intersection of two ( n2 − 1)dimensional subcones in Y x with vertices (TSX,z1 )∗ and (TSX,z2 )∗, so that Y12 is
an ( n2 − 2)-dimensional quadric). Thus (n − 2) − (α1 + α2 ) = n2 − 2, i.e.
α1 + α2 =
n
.
2
(3.3.2)
On the other hand, it is well known (cf. e.g. [37, Chapter XIII, § 4; 28, Chapter VI,
§ 1]) that the maximal dimension of linear £subspace
lying on a nonsingular n2 ¤
n
dimensional quadric Yi (i = 1, 2) is equal to 4 . Hence
hni
,
i = 1, 2.
(3.3.3)
αi ≤
4
3. DIMENSION OF SEVERI VARIETIES
81
Combining (3.3.2) and (3.3.3), we see that
hni n
α1 = α2 =
=
4
4
which simultaneously proves b) and c) (under specialization of z the dimension of
Yz ∩ Yi could only jump).
d) Let z ∈ S(C1 , C2 ) \ X. From b), c), and Proposition 3.1 it follows that the
set of quadrics Yu passing through x and intersecting Yz along a linear subspace of
dimension n4 is parametrized by the cone with vertex (TSX,z )∗ in Y x whose base is
a nonsingular ( n2 − 2)-dimensional quadric. For n > 4 this cone is irreducible, and
therefore all n4 -dimensional linear subspaces of the form Yz ∩ Yu belong to one and
the same family of linear subspaces on Yz . It is well known (cf. [37, Chapter XIII,
§ 4; 28, Chapter VI, § 1]) that the dimension of intersection of two n4 -dimensional
linear subspaces from one family on Y x has the same parity as n4 (we recall that on
n
the nonsingular even-dimensional quadric Yz2 there exist two irreducible families
of n4 -dimensional linear subspaces). On the other hand, in the notations used in
the proof of assertions b) and c)
Pα1 ∩ Pα2 = Y1 ∩ Y2 = x.
Hence for n > 4
n
(mod 2).
4
This completes the proof of assertion d).
¤
0≡
3.4. Remark. Assertions c) and d) of Proposition 3.3 were independently proved
by Fujita and Roberts (cf. Propositions 5.2 and 5.4 in [25]) who used the techniques
of computations with Chern classes. Their approach was developed by Roberts
(unpublished) and Tango [89] (cf. Remark 4.11 below).
3.5. Corollary. If in the conditions of Proposition 3.2 Yz1 6= Yz2 , then Yz1 ∩Yz2
is either a point or a linear subspace of dimension n4 .
In the proof of Proposition 3.3 we showed that varying z in S(C1 , C2 ) \ X we
obtain a family of n4 -dimensional linear subspaces on the n2 -dimensional quadric
Y1 . The base of this family is a nonsingular ( n2 − 2)-dimensional quadric. Hence
for n > 4 all linear subspaces of the form Yz ∩ Y1 , z ∈ S(C1 , C2 ) \ X belong to one
and the same irreducible family of n4 -dimensional linear subspaces on Y1 passing
through x (cf. the proof of Proposition 3.3 d)). We denote this family by F and the
other family by F 0 .
n
3.6. Lemma. Let n > 4, and let P04 be an arbitrary linear subspace on Y1
passing through x and belonging to the family F. Then for some z ∈ S(C1 , C2 ) \ X
we have Yz ∩ Y1 = P0 .
Proof. We argue by induction. Let z ∈ S(P0 , C2 ) \ X. Then Yz ∩ P0 is a linear
subspace of positive dimension. It is clear that it suffices to prove the following
assertion. Let
n
z ∈ S(C1 , C2 ) \ X,
Yz ∩ P0 = P α 3 x,
0<α<
4
82
IV. SEVERI VARIETIES
(since α ≡ n4 (mod 2), Proposition 3.3 d) shows that α is even). Then there exists
a point u ∈ S(C1 , C2 ) \ X such that Yu ∩ P0 ) P α .
Let y ∈ P0 \ P α . Then there exists a point y 0 ∈ Yz ∩ Y2 such that
hy, y 0 i 6⊂ X,
In fact, the variety
hy 0 , Pα i ⊂ Yz .
¯
Xz,y = {y 0 ∈ Yz ¯ hy, y 0 i ⊂ X}
is a linear subspace since otherwise there would exist a point
z 0 ∈ hy 0 , y 00 i \ X,
and it is clear that
hy, y 0 i ⊂ Yz 0 ,
y 0 , y 00 ∈ Xz,y ,
hy, y 00 i ⊂ Yz 0 ,
so that
y ∈ Yz 0 = Yz
(cf. Theorem 2.4 b) ) contrary to the choice of y. Since hy, P α i ⊂ P0 ⊂ X, we see
that P α ⊂ Xz,y and therefore
Xz,y 6= Yz ∩ Y2 .
¡
¢¡
¢
The linear subspace P α is contained in two 12 n4 − α n4 − α − 1 -dimensional
families of n4 -dimensional linear subspaces on Yz belonging to F and F 0 respecn
tively
(cf.¢ ¡[37, Chapter
¡
¢ XIII, § 4; 28, Chapter VI, § 1]). Since 4 and α are even,
1 n
n
2 4 −α
4 − α − 1 > 0. Hence there exists a linear subspace on Yz which passes
through P α and intersects Y2 at a point y 0 ∈ Y2 \ Xz,y . This point y 0 satisfies all
the above conditions.
Let
u ∈ hy, y 0 i \ X ⊂ S(C1 , C2 ) \ X,
and let a ∈ P α be an arbitrary point. By construction
hy, ai ⊂ P0 ⊂ X,
and therefore
hy, ai ⊂ Yu ,
Thus
hy 0 , ai ⊂ Yz ⊂ X,
hy 0 , ai ⊂ Yu .
P α ( hy, P α i ⊂ Yu .
¤
3.7. Remark. We can also take as P0 the n4 -dimensional linear subspace passing
through x and belonging to the family F 0 and repeat the arguments used in the
proof of Lemma 3.6 up to the last step when n4 −α = 1. In this case P α is contained
in exactly two linear subspaces, viz. the subspace Yz ∩ Y1 from the family F and
a subspace Pα 0 from the family F 0 . Since at that step the process of constructing
3. DIMENSION OF SEVERI VARIETIES
83
¡
¢
u must terminate, we see that if y ∈ P0 \ P α , then Xx,y = Pα0 and the n4 + 1 n
dimensional linear subspace hP0 , Pα0 i lies in X.
¡ n Thus
¢ each 4 -dimensional linear
0
subspace from the family F on Y1 is cut by an 4 + 1 -dimensional linear subspace
lying on X.
3.8. Remark. It is clear that for n = 4 each of the two lines making up C1 lies
on one of the quadrics Yz , z ∈ S(C1 , C2 ) \ X. These lines lie on certain planes in
X.
In the proof of Proposition 3.3 b) we already observed that the quadrics
Yz ,
z ∈ S(C1 , C2 ) \ X
¡n
¢
form a family parametrized by a nonsingular
−
2
2
¡n
¢ -dimensional quadric Y12 ,
x
where Y12 is the intersection in Y of the 2 − 1 -dimensional quadratic cones
∗
∗
with vertices (TSX,z1 ) and (TSX,z2 ) . On the other hand, the n4 -dimensional linear subspaces on the quadric Y1 passing through the point x and belonging to the
family F are parametrized by the spinor variety S x corresponding to the orbit of
highest weight vector of the spinor representation of the group Spin n2 (D n4 ), where
¡
¢
dim S x = 12 · n4 · n4 − 1 (cf. [11; 35; 87; 74]). The correspondence Yz à Yz ∩ Y1
induces a morphism ρ : Y12 → S x . Lemma 3.6 can now be restated as follows.
3.9. Corollary. For n > 4 the morphism ρ is surjective.
3.10. Theorem. Let X n ⊂ PN , n =
n = 2, 4, 8, or 16.
2
3 (N
− 2) be a Severi variety. Then
Proof. From Lemma 3.6 and Corollary 3.9 it follows that for n > 4
´
n
1 n ³n
− 2 = dim Y12 ≥ dim S = · ·
−1
2
2 4
4
(3.10.1)
(from Remark 3.8 it follows that for n = 4 the inequality (3.10.1) turns into equality). Thus for n > 2 we obtain the following inequality:
4 ≤ n ≤ 16.
(3.10.2)
From the definition of Severi varieties (cf. 1.2) it is clear that n is even, and so
Theorem 3.10 follows from (3.10.2) and Proposition 3.3 d). ¤
3.11. Remark. If n = 16, then S x is a six-dimensional quadric and ρ : Y12 → S x
is an isomorphism. For n = 8 we have S x = P1 , and ρ is the projection of twodimensional quadric onto one of its generatrices. In fact, for n = 8 each quadric
Yz ,
z ∈ S(P0 , C2 ) \ X
(3.11.1)
intersects Y1 along P0 , and thus each plane from the family F on Y1 is cut by a
pencil of quadrics from the family (3.11.1). Similarly, from Remark 3.7 it follows
that for n = 8 each plane from the family F 0 on Y1 is cut by a pencil of threedimensional linear subspaces (or, which is the same, by a four-dimensional linear
subspace) on X.
3.12. Remark. Tango [89] proved that if there exists a Severi variety X n, n > 16,
then n = 2m (m ≥ 7) or n = 3 · 2m (m ≥ 5).
84
IV. SEVERI VARIETIES
4. Classification theorems
Let z ∈ SX \ X be an arbitrary point. In this section we study the projection
πz : X 99K P n with center at the linear subspace Pz = SYz in more detail. We
already know (cf. Theorem 2.4 f)) that πz is an isomorphism outside the hyperplane
section Hz = X ∩ TSX,z .
¯
¡
¢
4.1. Lemma. For n > 2 the fibers of the map πz ¯Hz are n4 + 1 -dimensional
linear subspaces intersecting Yz along n4 -dimensional linear subspaces. For n > 4
these subspaces belong to the family F 0 , and for n = 4 the intersections contain
lines from both families.
Proof. Let
x ∈ Hz \ Yz ,
Pz,x = hPz , x i.
Suppose that x 0 ∈ (Hz \ Yz ) ∩ Pz,x , i.e. Pz,x0 = Pz,x . Then hx, x 0 i ⊂ X since
otherwise
u = hx, x 0 i ∩ Pz ∈ SX \ X
by Theorem 2.4 d) and therefore x ∈ Yu = Yz contrary to the choice of x (cf. the
proof of assertion e) of Theorem 2.4). As in the proof of Lemma 3.6, from this it
follows that
¢
¡
πz−1 πz (x) = (Pz,x ∩ X) \ Yz
is a linear subspace. The intersection of this linear subspace with the quadric Yz
coincides with the linear subspace Xz,x ⊂ Yz introduced in the proof of Lemma 3.6.
The assertion of Lemma 4.1 now follows from Remarks 3.7 and 3.8. ¤
Thus for n > 2 Bz = πz (Hz ) is an [(n − 1) − ( n4 + 1)] = ( 3n
4 − 2)-dimensional
subvariety in the hyperplane in P n corresponding to the hyperplane TSX,z ⊂ PN .
4.2. Lemma. a) If n = 4, then Bz is a union of two skew lines in P3 ⊂ P4 .
b) If n = 8, then Bz is the Segre variety P1 × P3 ⊂ P7 ⊂ P8 .
c) If n = 16, then Bz ⊂ P15 ⊂ P16 is the spinor variety corresponding to
the orbit of highest weight vector of the spinor representation of the group
Spin10 (D5 ) (cf. § 2 of Chapter III).
Proof. Let Sz be the variety parametrizing the n4 -dimensional linear subspaces
on Yz belonging to the family F 0 . Then Sz corresponds to the orbit of highest
weight vector of the spinor representation of the group Spin n2 +2 (D n4 +1 ) and
dim Sz =
´
1 n ³n
· ·
+1
2 4
4
(cf. [11; 35; 87; 37, Chapter XIII, § 4]). By Lemma 4.1, the correspondence
a à πz−1 (a) ∩ Yz
induces a morphism
Bz → Sz .
(4.2.1)
4. CLASSIFICATION THEOREMS
85
From Remark 3.7 it follows that this morphism is surjective.
In case c) the varieties Bz and Sz have the same dimension (dim Bz = dim Sz =
10) and the map (4.2.1) is an isomorphism. In fact, if
x, x 0 ∈ Hz \ Yz ,
Pz,x 6= Pz,x0 ,
Pz,x ∩ Yz = Pz,x0 ∩ Yz ,
then for each u ∈ S(Pz,x , Pz,x0 ) \ X
Yu ⊃ Pz,x , Pz,x0
which is impossible since Yu is a nonsingular n2 -dimensional quadric and dim Pz,x =
n
0
4 + 1. Therefore S(Pz,x , Pz,x ) ⊂ X and the fiber of the map (4.2.1) over the point
corresponding to Pz,x ∩ Yz is a linear subspace of positive dimension which is also
15
would contain an exceptional
impossible since otherwise the variety A10
z ⊂ P
divisor contrary to a theorem of Barth (cf. [6; 33]). Thus in case c) the map (4.2.1)
is an isomorphism, and from the fact that
Pic Sz ' Z
(cf. [11; 35; 87]; by virtue of Theorem 5 from [95] this also follows from the results
of § 2 of Chapter III, and in the case when dim Sz = 10 one can apply a Barth type
theorem [54; 65; 60]) it follows that the embedding
15
A10
,→ P16
z ,→ P
corresponds to the spinor representation.
In case b)
dim Bz = 4,
Sz = P 3 .
From Remark 3.11 it follows that the fibers of the morphism (4.2.1) are projective
lines and the preimage of an arbitrary projective line from Sz (corresponding to
a point from Yz ) is a nonsingular two-dimensional quadric. Furthermore, to each
point of Sz there corresponds a four-dimensional linear subspace on X mapping
to a line on Bz , and thus we obtain a map Sz × P1 → Bz . Thus in case b) Bz is
projectively isomorphic to the Segre embedding of the variety P1 × P3 in P7 ⊂ P8 .
In case a)
dim Bz = 1,
Sz = P 1 .
By Remark 3.8, each
` line on the quadric Yz is cut by a plane, and`we obtain a
surjection Bz → Sz Sz . Arguing as in case c), we see that Bz = P1 P1 and Hz
consists of two irreducible components intersecting along Yz .
¤
4.3. Remark. Since
TSX,z ∩ SX = T (Yz , X),
we see that
Hz = TSX,z ∩ X =
[
y∈Yz
TX,y ∩ X.
86
IV. SEVERI VARIETIES
Furthermore,
TX,y ∩ X =
[
Cu =
[
n
y∈P 4
u∈Sy XrX
n
P4,
Cu = Yu ∩ TYu ,y .
⊂X
Hence TX,y ∩ X is a cone with vertex y, and from Proposition 3.1 and Corollary 3.5
it follows that
´ n
n ³n
3n
dim (TX,y ∩ X) = +
−1 − =
− 1.
2
2
4
4
Under the mapping πz each of the cones TX,y ∩ X is projected onto its base which
is isomorphic to the variety Bz .
4.4. Remark. For n = 2
Hz = 2Yz ,
TX,y ∩ X = y
(by Corollary 1.15 from Chapter I, the hyperplane section Hz is reduced for n > 2
and is normal for n > 4). It is not hard to show that in this classical case the map
πz is an isomorphism, so that Bz = ∅ (cf. e.g. [82; 62]).
Next we describe the rational map inverse to the map πz : X 99K P n . This map
σz is defined by 3n
2 + 3 forms
G0 , . . . , G 3n
,
2 +2
deg Gi = d,
i = 0, . . . ,
3n
+2
2
vanishing on a subvariety Bz ⊂ Pn−1 ⊂ P n . It is easy to see that Bz is defined in Pn−1 (and in P n ) by quadratic equations. This immediately follows from
Lemma 4.2, but can also be proved without any computations. It suffices to observe that, according to Remark 4.3, Bz is the base of the cone TX,y ∩ X (y ∈ Yz ).
Hence Bz = Pn−1 ∩ X, where Pn−1 is a hyperplane in TX,y not passing through
y. In Remark 2.5 we observed that X is defined by quadratic equations. Restricting these equations on Pn−1 we obtain equations for Bz . It is clear that the
image of the restriction of the map κX from Remark 2.5 (cf. (2.5.3)) on TX,y (or
Pn−1 ⊂ TX,y ) coincides with the n2 -dimensional quadric Y x from Proposition 3.1.
Hence the number of linearly independent quadratic equations defining Bz in Pn−1
is equal to n2 + 2. Adding n + 1 quadratic equations defining Pn−1 in P n , we see
that the subvariety Bz ⊂ P n is defined by 3n
2 + 3 linearly independent quadratic
equations. From this it follows that for n > 2 we have d = 2. The case n = 2
(Bz = ∅) is dealt with in a similar way (cf. Remark 4.4); this case was first studied
by Severi (cf. [82; 62; 15]). Summing up, we obtain the following result.
4.5. Theorem. If X n ⊂ PN , n = 23 (N − 2) is a Severi variety, then n = 2, 4,
8, or 16 and X is the image of P n under the rational map σ : P n 99K PN defined by
the linear system of quadrics passing through a subvariety A ⊂ Pn−1 ⊂ P n , where
a) for n = 2
A = ∅; `
b) for n = 4
A = P1 P1 is a union of two skew lines;
c) for n = 8
A = P1 × P3 is the Segre variety in P7 ;
d) for n = 16
A = S 10 is the spinor variety parametrizing one of the
two families of four-dimensional linear subspaces on the
nonsingular quadric in P9 (cf. § 2 of Chapter III).
4. CLASSIFICATION THEOREMS
87
In other words, the Severi variety X n is obtained from the Veronese variety
n(n+3)
v2 (P n ) ⊂ P 2
(n = 2, 4, 8, 16) by projecting it from the linear span hv2 (A)i of
the image of the subvariety A ⊂ P n under the Veronese embedding v2 .
4.6. Remark. From Remark 4.3 and the arguments given before the statement
of Theorem 4.5 it immediately follows that the linear system of quadrics cut in
a general linear subspace Pn−1 ⊂ TX,x by the linear system of quadrics passing
through X and defining a rational map
n
n
Pn−1 99K Q 2 ⊂ P 2 +1
n
(where Q 2 = Y x is a nonsingular quadric) is the second fundamental form in the
sense of [29] and the subvariety A ⊂ Pn−1 is the fundamental subset of this form.
Theorem 4.5 shows that in each of the dimensions 2, 4, 8, 16 there exists at most
one Severi variety. To complete classification of Severi varieties it remains to verify
that the necessary conditions formulated in Theorem 4.5 are also sufficient, i.e. the
varieties X n described in Theorem 4.5 are nonsingular and can be isomorphically
3n
projected to P 2 +1 .
However in Chapter III we already constructed four examples of Severi varieties
(the first three of them, viz. the Veronese, Segre, and Grassmann varieties, are
classical; cf. Remark 1.3, [33; 38]). Moreover, using methods from representation
theory, in § 3 of Chapter III we studied the maps πz and σz in these examples
and described geometric properties and computed invariants of the corresponding
varieties. Thus Theorem 4.5 yields the following basic result.
4.7. Theorem. Over an algebraically closed field of characteristic zero each
Severi variety is projectively equivalent to one of the following four projective varieties:
a)
b)
c)
d)
v2 (P2 ) ⊂ P5
P2 × P2 ⊂ P8
G(5, 1)8 ⊂ P14
E 16 ⊂ P26
(Veronese surface);
(Segre variety);
(Grassmann variety);
(Cartan variety);
All these varieties are homogeneous, rational, and are defined by quadratic equations. Furthermore, Severi variety X corresponds to the orbit of highest weight
vector of an irreducible representation of a semisimple group G in a vector space V
with highest weight Λ, where:
a)
b)
c)
d)
G = SL3 ,
G = SL3 × SL3 ,
G = SL6 ,
G = E6 ,
Λ = 2ϕ1 ;
Λ = ϕ1 ⊕ ϕ1 ;
Λ = ϕ2 ;
Λ = ϕ1
(here ϕi is the i-th fundamental weight).
¯
The variety SX corresponds to the cone of ‘null-forms’ N = {v ∈ V ¯ F (v) = 0},
where F is the cubic form generating the algebra of G-invariant polynomials on V .
There remains one question: how to ‘explain’ the fact that dimension of Severi
varieties assumes only these four values (or at least that they are powers of two)?
One approach suggested by Roberts (unpublished) and Tango [89] is based on a
88
IV. SEVERI VARIETIES
study of arithmetic properties of Chern characters (Roberts used well known number theoretic results on denominators of Bernoulli numbers). Using this approach
Tango showed that if there exists a Severi variety X n of dimension n > 16, then
n = 2m (m ≥ 7) or n = 3 · 2m (m ≥ 5). More intriguing is the ‘explanation’ based
on the following result (independently discovered by Roberts).
4.8. Theorem. Let A be a composition algebra over the field K, and let J be
the Jordan algebra of Hermitean 3×3 -matrices over A (a matrix A is called Hermitean if Āt = A, where t denotes transposition and the bar denotes the involution
n
N
in A), so that dimK J = 3(dimK A+1) (cf. [10; 44; 76]).
¯ Let X ⊂ P(J) = P be the
projective variety corresponding to the cone {A ∈ J ¯ rk A ≤ 1}. Then ¯X is a Severi
variety and SX is the hypersurface corresponding to the cone {A ∈ J ¯ det A = 0}.
Conversely, each Severi variety is obtained in such way.
Proof. By Jacobson’s theorem (cf. [43; 44, Chapter IV, n0 3]), there exist exactly
four composition algebras—one in each of the dimensions 1, 2, 4, 8, viz. the algebras
A0 = K, A1 = K[t]/(t2 + 1), A2 —the algebra of quaternions over K, and A3 —the
Cayley algebra over K. For these algebras
Ni = dim P(Ji ) = 3 · 2i + 2,
ni = dim Xi = 2i+1 = 2 dim Ai ,
where Ji and Xi are the Jordan algebra and the projective variety corresponding
to the algebra Ai (0 ≤ i ≤ 3).
It is clear that the surface X0 coincides with the Veronese surface. The algebra
J1 is identified with the algebra of 3×3 -matrices over the field K, and the variety X1
is identified with the Segre variety P2 × P2 (cf. Theorem 2.4 in Chapter III). Since
the field K is algebraically closed, A2 is isomorphic to the algebra of 2×2 -matrices
over K. From this it is easy to deduce that X2 is projectively equivalent to G(5, 1)
(cf. Chapter III, 2.5, A3 ) ). Finally, from Freudenthal’s results [23] it follows that
the variety X3 is isomorphic to E (cf. Chapter III, 2.5, E) ). ¤
It is easy to see that Theorem 4.8 can be restated as follows (judging by [56], a
similar result (for complexifications of real division algebras) was proved by T. Banchoff).
4.9. Theorem. X is a Severi variety if and only if X is a ‘Veronese surface’
over one of the algebras Ai (0 ≤ i ≤ 3), i.e. X is the image of the ‘projective plane’
P2 (Ai ) = (A3i \ 0)/A∗i (where A∗i is the set of invertible elements of the algebra Ai )
with respect to the map
(x0 : x1 : x2 ) 799K (· · · : xl x̄m : · · · ),
0 ≤ l ≤ m ≤ 2.
4.10. Remark. We do not know if there exists some intrinsic connection between
composition algebras (or some other class of algebras) and Severi varieties or this
is an accidental coincidence. In any case, classification of Severi varieties given in
Theorems 4.7–4.9 allows to give a new unexpected proof of the well known Jacobson
theorem on the structure of composition algebras (cf. e.g. [43; 44, Chapter IV, n0 3]).
In Chapter VI (cf. Remark 5.10 and Theorem 5.11) we shall see that extremal
varieties with small secant varieties also correspond to matrix Jordan algebras (or
4. CLASSIFICATION THEOREMS
89
to Veronese varieties over composition algebras). Furthermore, all varieties except
E 16 ⊂ P26 correspond to special Jordan algebras, and the variety E corresponds to
the exceptional algebra of Hermitean 3×3 -matrices over the Cayley numbers.
4.11. Remark. In the case of surfaces Theorem 4.7 was first proved by Severi [82]
(the proof of Severi is reproduced in [62], and the paper [15] is devoted to finding out
which parts of this proof work in the case when char K > 0). Griffiths and Harris
who apparently didn’t know about Severi’s paper proved a local version of his result
(cf. [29, 6c)]). Scorza [77; 78] classified (possibly singular) threefolds and fourfolds
with small secant varieties. However these results of Scorza were forgotten, and
in 1979 Griffiths and Harris proved that each four-dimensional Severi variety (or a
Zariski open subset of such a variety) has the same second fundamental form as the
Segre variety P2 × P2 ⊂ P8 (cf. [29, (5.62)]). In the same paper Griffiths and Harris
conjectured that up to projective equivalence P2 × P2 is the only four-dimensional
Severi variety. Basing on the author’s results, Fujita and Roberts [25] proved this
conjecture, and Fujita [24] gave a modern proof of Scorza’s result for nonsingular
threefolds. Tango [89] showed that if there exists a Severi variety X n, n > 16, then
n ≥ 96 and either n = 2m or n = 3 · 2m, where m is a natural number. Theorem 4.7
was first proved in [99] (cf. also [56]).
90
IV. SEVERI VARIETIES
5. Varieties of codegree three
From Theorem 2.4 and Remark 2.5 it follows that the dual variety of an arbitrary Severi variety is a cubic hypersurface. This property is shared by isomorphic
projections of Severi varieties. This observation indicates that in the context of the
present chapter it is relevant to give classification of all nonsingular varieties whose
dual varieties have degree three.
Classification of varieties of small degree has been a popular topic since A. Weil
[104] classified all projective varieties of degree three in 1957 (classification of varieties of degree one and two is trivial). Later Swinnerton-Dyer [86] succeeded
in classifying varieties of degree four, and in a series of papers Ionescu classified
smooth projective varieties up to degree eight. Several papers are devoted to lowdimensional varieties of small degree and to varieties whose degree is not too big
with respect to codimension. Due to efforts of Hartshorne, Barth, Van de Ven, and
Ran it was found out that if the dimension of a nonsingular variety is sufficiently
large with respect to its degree, then the variety is a complete intersection.
However here we are more interested in class which is another classical invariant
of projective varieties whose role in enumerative geometry is not less than that of
degree. Traditionally, the class of a nonsingular variety X n ⊂ PN is defined as the
number of singular divisors in a generic pencil of hyperplane sections of X. Thus
if the dual variety X ∗ is a hypersurface, then the class of X is equal to the degree
of X ∗ . Varieties for which codim X ∗ > 1 have class zero, but for us it is more
convenient to use the notion of codegree.
5.1. Definition. The number d∗ = deg X ∗ is called the codegree of X in PN
and is denoted by codeg X.
Thus codegree is equal to class provided that X ∗ is a hypersurface.
It is clear that the only varieties of codegree one are linear subspaces P n ⊂ PN
and the only varieties of codegree two are quadrics Qn ⊂ PN .
We notice that if X n ⊂ PM ⊂ PN , then the dual variety of X in PN is the cone
over the dual variety of X in PM with vertex (PM )∗ = PN −M −1 . Hence the codegree
of X in PN is equal to the codegree of X in PM , and in classification of varieties
of a given codegree it suffices to consider the case when X is nondegenerate, i.e.
hXi = PN , where hXi is the linear span of X.
Classification of nonsingular varieties of small codegree is apparently more difficult than that of varieties of small degree, e.g. because in the last case one can
proceed by induction on dimension by taking hyperplane sections while in the first
case there is no such possibility. Furthermore, the flavor of the problem for codegree is quite different. An important part of the problem is to characterize the
structure of singularities of hypersurfaces of a given degree whose dual varieties
are nonsingular. While there always exist varieties of a given degree and arbitrary
dimension (e.g. hypersurfaces), there are reasons to expect that, if we denote by
n(d) the smallest natural number (or ∞) such that for each nonsingular variety X
with codeg X = d we have dim X ≤ n(d), then n(d) < ∞ for d > 2 (of course,
n(2) = ∞). However the number n(d) is not small; in the present section we show
that already n(3) = 16.
5. VARIETIES OF CODEGREE THREE
91
There are many papers devoted to surfaces of small class (codegree) and some
papers devoted to threefolds (cf. [53]; a survey and bibliography can be found in
[91]), but in general varieties of small codegree remain completely unexplored. In
the present section we make the first step and give complete classification of nonsingular nondegenerate varieties of codegree three (it turns out that up to projective
equivalence there are exactly ten such varieties).
5.2. Theorem. Let X n ⊂ PN be a nonsingular irreducible nondegenerate
projective variety of codegree three over an afslgebraically closed field K of characteristic zero. Then there are the following possibilities:
0. n = 3, X = P1 × P2 ⊂ P5 (X is a Segre variety);
I. n = 2, X = F1 ⊂ P4 (F1 is a bundle with fiber P1 over P1 embedded
in P4 so that its fibers and the minimal section s are projective lines and
(s2 ) = −1);
II. X is a Severi variety. More precisely, in this case there are the following
possibilities:
II.1. n = 2, X = v2 (P2 ) ⊂ P5 (X is the Veronese surface);
II.2. n = 4, X = P2 × P2 ⊂ P8 (X is the Segre variety);
II.3. n = 8, X = G(5, 1) ⊂ P14 (X is the Grassmann variety);
II.4. n = 16, X = E ⊂ P26 (X corresponds to the orbit of highest weight
vector of the nontrivial representation of the group E6 having the
smallest possible dimension);
3n
0
II . X is an isomorphic projection of one of the Severi varieties X n ⊂ P 2 +2
3n
described in II to P 2 +1 , n = 2i , 1 < i < 4 (as in II, here we obtain four
cases II0.1– II0.4).
5.3. Remark. In all the above cases X is the image of Pn under the rational
map defined by the linear system of quadrics in P n passing through B, where B
has the following form:
`
0. B = P0 P1 ;
I. B = P0 ;
II.1, II0.1. B = ∅;
`
II.2, II0.2. B = P1 P1 ;
II.3, II0.3. B = P1 × P3 ;
II.4, II0.4. B = S 10 , the spinor variety in P15 corresponding to the orbit of
highest weight vector of the spinor representation of the group Spin9
(or Spin10 ).
In case 0 we have X ∗ ' X ' P1 × P2 ; in case I X ∗ is the projection of P1 × P2
from a point of P5 \ P1 × P2 . If X is a Severi variety, then by (2.5.4) X ∗ ' SX,
and in case II0 the variety X ∗ is obtained from the corresponding Severi variety by
intersecting it with a general hyperplane.
According to Theorems 4.8 and 4.9, all Severi varieties can be interpreted as
‘matrices of rank 1’ in the space of Hermitean 3 × 3 -matrices (or as ‘Veronese
surfaces’ ) over one of the four standard composition algebras; X ∗ is defined by the
equation det = 0 and therefore has degree three. In case II0 X ∗ is defined by the
same equation in the subspace of matrices with vanishing trace.
92
IV. SEVERI VARIETIES
Variety I is a hyperplane section of variety 0; these varieties have degree three.
Varieties II.1 and II0.1 have degree four, varieties II.2 and II0.2 have degree six,
varieties II.3 and II0.3 have degree 14, and varieties II.4 and II0.4 have degree 78
(cf. Chapter III, Proposition 2.10).
The remaining part of this section is devoted to a proof of Theorem 5.2.
5.4. Lemma. In the conditions of the theorem, let Σk = Sing X ∗ . Then
SΣ ⊂ X ∗ .
Proof. For α ∈ Σ we have multα X ∗ ≥ 2. Hence if α, β ∈ Σ, α 6= β, then the
line hα, βi intersects X ∗ with multiplicity at least 4, and therefore this line lies in
X ∗ . Thus SΣ ⊂ X ∗ . ¤
5.5. Remark. Since codim X ∗ ≤ deg X ∗ − 1 = 2, there are two possibilities:
codim X ∗ = 1 and codim X ∗ = 2. The second case is easy to investigate since
classification of varieties of degree 3 and codimension 2 is fairly simple: all nondegenerate varieties with such invariants are cones over sections of the Segre variety
P1 × P2 ⊂ P5 by linear subspaces of P5 (cf. [104]), and since X is nondegenerate
and codim X > 1 (because otherwise codeg X = deg X · (deg X − 1)dim X 6= 3), X
is the Segre threefold 0. One can avoid reference to [104] by considering a general
hyperplane section Y of a variety X with codim X ∗ = 2. It is easy to see that
Y ∗ ⊂ P(N −1)∗ is obtained by projecting X ∗ from a general point of PN ∗ . Since
Y ∗ is a hypersurface, it suffices to classify varieties X of codegree 3 for which X ∗
is a hypersurface and to find out which of them are smoothly extendible, i.e. are
hyperplane sections of nonsingular varieties (cf. Corollaries 5.7 and 5.10).
Thus in what follows we may assume that X ∗ is a hypersurface.
5.6. Lemma. Either Σ = PN −2 or the hypersurface X ∗ is normal.
Proof. Suppose that X ∗ is not normal. Then from the Serre normality criterion
it follows that there exists a component Σ0 ⊂ Σ such that dim Σ0 = dim X ∗ − 1 =
N − 2, and Lemma 5.4 shows that SΣ0 ⊂ X ∗ . Let Λ be a general plane in PN ∗ ,
and put X ∗ 0 = X ∗ ∩ Λ, Σ00 = Σ0 ∩ Λ. Then X ∗ 0 is an irreducible plane cubic,
Σ00 is a union of deg Σ0 distinct points and SΣ00 ⊂ X ∗ 0 . Hence deg Σ0 = 1 and
Σ0 = PN −2 . Furthermore, Σ = Σ0 since otherwise from Lemma 5.4 it would follow
that X ∗ contains the hyperplane spanned by Σ0 and a point from Σ \ Σ0 . ¤
5.7. Corollary. Let X n ⊂ PN be a nonsingular projective variety such that
codeg X = 3, codim X ∗ = 2, and let Y n−1 ⊂ PN −1 be a general hyperplane section
of X. Then codeg Y = 3, codim Y ∗ = 1 and Sing Y ∗ = PN −3 .
Proof. It is clear that Y ∗ is obtained by projecting X ∗ ⊂ PN ∗ from a general
point ξ ∈ PN ∗ . Since SX ∗ = PN ∗ (to prove this it suffices to consider the section
of X ∗ by a general three-dimensional linear subspace Λ ⊂ PN ∗ ), the finite map
X ∗ → Y ∗ is not an isomorphism. Hence the variety Y ∗ cannot be normal, and
from Lemma 5.6 it follows that Sing Y ∗ = PN −3 . ¤
Since X ∗ is a hypersurface, α ∈ Sm X ∗ if and only if the hyperplane section
Lα ∩ X has a unique nondegenerate quadratic singular point (cf. [16]). For x ∈ X
we denote by Σx ⊂ Px the set of hyperplanes β ∈ Px for which the singularity of the
hyperplane section Lβ ∩ X at the point x is not a nondegenerate quadratic singular
5. VARIETIES OF CODEGREE THREE
93
N∗
point (we Ã
identify P
!x and Σx with their images
à in P ! under the morphism π).
[
[
Then Σ ⊃
Σx and the points from Σ \
Σx correspond to hyperplane
x∈X
x∈X
sections having several nondegenerate
quadratic singular points. In the case when
S
X ∗ is normal we have Σ =
Σx .
x∈X
Let x ∈ X be a point for which Σx 6= Px . Taking N −n points α0 , . . . , αN −n−1 in
general position in Px and denoting by Aα , α ∈ Px the quadratic term in the Taylor
expansion of the equation of the hyperplane section Lα ∩ X in some system of local
coordinates in a neighborhood
of the point x in X,¯we see that Σx ⊂ Px is defined
¯
by the equation det ¯t0 Aα0 + · · · + tN −n−1 AαN −n−1 ¯ = 0, and so for N ≥ n + 2 Σx
is a hypersurface of degree at most n in Px = PN −n−1 . Since Σ ⊂ PN ∗ is defined by
quadratic equations (by vanishing of the partial derivatives of the equation of X ∗ ),
for all x for which Σx is distinct from Px the subvariety Σx ⊂ Px is a hypersurface
of degree at most two.
For N ≥ n + 2 we will distinguish between the following two main cases:
I. For all x ∈ X for which Px 6⊂ Σ the subvariety Σx is a hyperplane in Px ;
II. For a general x ∈ X, Σx is a quadric in Px .
For the sake of completeness this list can be supplemented by the following case
which occurs if and only if dim X ∗ < N − 1:
0. Σx = Px for all x ∈ X.
5.8. Lemma. Under the above assumptions, case I occurs iff Σ is a linear
subspace of PN , and case II occurs iff SΣ = X ∗ .
Proof. If for a general point x ∈ X the subvariety Σx is a quadric, then SΣx = Px
and therefore SΣ = X ∗ .
If Σ is a linear subspace, then Σx ⊂ Σ∩Px , and if Px 6⊂ Σ, then Σx is a hyperplane
in Px . It remains to show that in case I Σ is a linear subspace. SupposeSthat this
Σx and
is not so. By Lemma 5.6 we may assume that X ∗ is normal so that Σ =
x∈X
for each x ∈ X Σx = Σ ∩ Px is a linear subspace of Px (of codimension 0 or 1). Let
α, β be a generic pair of points of Σ, and let l = hα, βi 6⊂ Σ. Consider the family
of planes Πt ⊂ PN ∗ passing through l (here t runs through an (N − 2)-dimensional
linear space of parameters). Then Πt ∩ X ∗ = Ct , where Ct is a curve of degree
three, Ct = l + Qt . By definition, Ct is singular at the points α and β, so that
for all t Qt 3 α, β. Since l 6⊂ Σ = Sing X ∗ , for a general point γ ∈ l the tangent
space TX ∗ ,γ is a hyperplane in PN ∗ . Furthermore, the hyperplane TX ∗ ,γ ⊃ l is nonconstant
¡ when¢ γ runs
¡ through ¢l \ Σ since otherwise we would have l ⊂ Py , where
y = p π −1 (γ) = p π −1 (l \ Σ) , and since α, β ∈ Σy and Σy is a linear subspace
S
of Py , l ⊂ Σy ⊂ Σ contrary to the choice of l. Thus
TX ∗ ,γ = PN ∗ and for a
γ∈l\Σ
general plane Πt ⊃ l there exists a point γ ∈ l \ Σ such that Πt ⊂ TX ∗ ,γ , i.e. Ct
is singular at the point γ and (Qt ∩ l) 3 α, β, γ. Since Qt is a conic, from this it
follows that Qt ⊃ l so that Πt ⊂ TX ∗ ,γ for all γ ∈ l which is clearly impossible
for generic t. This contradiction shows that l ⊂ Σ, i.e. SΣ = Σ and Σ is a linear
subspace of PN ∗ . ¤
94
IV. SEVERI VARIETIES
We begin with investigating case I.
5.9. Lemma. In case I, X = F1 is a rational scroll of degree three (the image
of P2 under the rational map defined by the linear system of conics passing through
a fixed point of P2 ).
Proof. If X n is a hypersurface of degree m, then deg X ∗ = m(m − 1)n and
so codeg X 6= 3. Therefore codim X ≥ 2. From Lemma 5.8 it follows that for
each point x ∈ X the subvariety Σ ∩ Px is a linear subspace of Px . Let x be a
generalSpoint of X. Then Σx = Px ∩ Σ = PN −n−2 is a hyperplane in Px . Let
SX = Σx ⊂ PX , where x runs through the set of general points of X, so that SX
x
is a divisor in PX . Then π(SX ) = Σ0 ⊂ Σ, and for a general point x ∈ X we have
Σx = Px ∩ Σ = Px ∩ Σ0 = Σ0x .
Let H be a hyperplane section of X ∗ passing through Σ. Then π ∗ (H) = SX +PD ,
where D is an effective divisor in X and PD = p−1 (D) is a divisor in PX . For an
arbitrary point α ∈ Σ we denote by Yα both the preimage of α in PX and the image
of this preimage in X. By the projection formula, for α ∈ Σ \ Σ0 we have
¡
¢
¡
¢
¡
¢
0 = Yα · π ∗ (H) P = Yα · PD P = Yα · D X .
X
X
We notice that if k = dim Σ < N − 2, then dim Yα > 0 since in this case the
hypersurface X ∗ is normal. Furthermore, if N ≤ 2n − 2, then Pic X = Z (cf. [54]),
so that D is a positive multiple of the hyperplane section of X and the equality
(Yα · D) = 0 is impossible.
We claim that for k ≤ N − 5 X can be isomorphically projected to P2n−2 . In
fact, for a generic pair of points x, y ∈ X we have
dim Px ∩ Py ≥ dim Σ0x ∩ Σ0y ≥ dim Σ0x + dim Σ0y − dim Σ = 2(N − n − 2) − k,
so that
dim hTx , Ty i ≤ N − 2(N − n − 2) + k − 1 = 2n + 3 − (N − k).
By Terracini’s lemma (cf. Theorem 1.13 in Chapter II), X can be isomorphically
projected to PdimhTx ,Ty i . Hence for k ≤ N − 5 X can be isomorphically projected
to P2n−2 . We have already shown that for N ≤ 2n − 2 there are no varieties of
codegree three. Since codegree is stable with respect to general projections, such
varieties also do not exist for 0 ≤ k ≤ N − 5.
It remains to consider the case when k ≥ N − 4. We observe that Σ0 6= Σ since
otherwise from the Bertini theorem it would follow that X = p(SX ) ⊂ L, where
L = (Σ)∗ = PN −k−1 contrary to the assumption that X is nondegenerate. It is
N −2
clear that π −1 (Σ) = SX
∪ PE , where E is a subvariety of X, PE = p−1 (E).
Suppose that k = N − 3. Then dim PE − dim Σ = dim E − n + 2 ≥ 1, i.e.
dim E = n − 1. When α runs through the set of general points of Σ, the varieties
Yα sweep out a dense subset in E, and by Bertini’s theorem E n−1 ⊂ L = (Σ)∗ = P2 ,
so that n ≤ 3. For β ∈ Σ0 we have dim Yβ ≥ dim SX − (k − 1) = 2. Hence n = 3
and for β ∈ Σ0 the hyperplane section (β)∗ · X is not reduced. In particular, for
a general point x ∈ X a general hyperplane section from Σx is not reduced at x,
5. VARIETIES OF CODEGREE THREE
95
and from the Bertini theorem it follows that the linear subspace (Σx )∗ = P4 is
tangent to X along a surface in contradiction with the theorem on tangencies (cf.
Corollary 1.8 in Chapter I). Thus the case k = N − 3 is impossible.
Suppose now that k = N − 4. As we already observed, in this case X can be
isomorphically projected to P2n−1 , and without loss of generality we may assume
that N = 2n − 1. For β ∈ Σ0 we have dim Yβ ≥ dim SX − (k − 1) = 3, and from
the theorem on tangencies it follows that n > 4.
Since dim PE − dim Σ = dim E − n + 3 ≥ 1, we have dim E ≥ n − 2, and
arguing as in the case k = N − 3 we see that E ⊂ L = (Σ)∗ = P3 . Hence the
case dim E = n − 1 is impossible, and for dim E = n − 2 we have n = 5, N = 9
and E = L = P3 . Moreover, from the above it follows that in the last case Σ0 is a
hypersurface in Σ = P5 and dim Yβ = 3 for a general point β ∈ Σ0 .
S
Let x be a general point of X, and let Yx = Yβ , where β runs through the
β
set of general points of Σx . By the theorem on tangencies, dim Yx ≥ 4. On the
other hand, if dim Yx = 5, then Yx = X so that for a general point y ∈ X there
exists a hyperplane (β)∗ which is tangent to X at x and y. From the Terracini
lemma it follows that X can be isomorphically projected to P8 which was already
shown to be impossible (8 = 2n − 2). Thus dim Yx = 4, and Bertini’s theorem
¢6
¡
5
, L3 i. Let y be a general point of Yx .
yields the inclusion Yx ⊂ (Σx )∗ = hTX,x
A dimension count shows that y lies on a one-dimensional family of Yβ . Hence
Σx ∩ Σy = Px ∩ Py = P1 , and a dimension count shows that, varying y ∈ Yx ,
we thus obtain a S
general line in the plane Σx . From the theorem on tangencies it
follows that dim Yγ = 4, where γ runs through the set of general points of the
γ
S
line Σx ∩ Σy . Thus Yγ coincides with both Yx and Yy , so that Yy = Yx and
γ
Yx ⊂
\
y
hTX,y , Li =
­[ ®∗
Σy ,
y
where y runs through the set of general points of Yx .
Since for a general point y ∈ Yx we have dim Σx ∩ Σy = 1, dim hΣx , Σy i∗ = 5
and Yx4 ⊂ hΣx , Σy i∗ = P5x ⊃ L3 . Let x 0 be another general point of X. Then
Yx0 ⊂ P5x0 ⊃ L3 and dim hP5x , P5x0 i ≤ 10 − 3 = 7. Since Yx ∩ Yx0 is nonempty (these
subvarieties intersect with each other on L), we have dim Yx ∩ Yx0 ≥ 8 − 5 = 3.
Thus the linear subspace hP5x , P5x0 i is tangent to X along the subvariety Yx ∩ Yx0
which contradicts the theorem on tangencies since
dim Yx ∩ Yx0 ≥ 3 > 2 ≥ dim hP5x , P5x0 i − dim X.
Thus the case k = N − 4 is also impossible.
It remains to consider the case k = N − 2. Since Σ = π(PE ), we have dim PE =
dim E + (N − n − 1) > dim Σ = N − 2, and so dim E = n − 1. Arguing as in the
case k = N − 3, we see that E n−1 ⊂ L = (Σ)∗ = P1 and therefore n ≤ 2. Since for
β ∈ Σ0 we have dim Yβ ≥ (N − 2) − (k − 1) = 1, X is a surface and E = L is a line
on X. Now the lemma follows from Proposition 3 from [96]. ¤
96
IV. SEVERI VARIETIES
5.10. Corollary. Let X n ⊂ PN be a nondegenerate nonsingular variety such
that codeg X = 3, codim X ∗ = 2. Then n = 3, N = 5 and X = P1 × P2 ⊂ P5 is a
Segre variety.
Proof. From Corollary 5.7 it follows that a general hyperplane section Y of the
variety X satisfies the conditions of Lemma 5.9, and so Y = F1 . The corollary
now follows from well known results on extension of projective varieties (in this
particular case it is easy to verify directly that the standard morphisms F1 → P1
and F1 → P2 defined by the linear systems |F | and |s + F | respectively extend to
∼
X and define an isomorphism X → P1 × P2 ). ¤
A different proof of the corollary is given in Remark 5.5.
5.11. Remark. A close analysis of our proof of Lemma 5.9 shows that we actually
used only the fact that Σ = Sing X ∗ is a linear subspace. Thus our method allows
to give classification of all varieties having this property (the list of such varieties
includes all rational scrolls Fe (e ≥ 0) of degree e + 2 embedded in PN (N ≤ e + 3)
by means
¡ of¢ a very ample linear subsystem of the linear system |s+(e+1)F |, where
s ' P1 , s2 X = −e is the minimal section and F ' P1 is a fiber). Similarly, using
our techniques it should be possible to classify all varieties for which SΣ ⊂ X ∗ (cf.
Lemma 5.4).
To prove the theorem it remains to consider case II. We begin with giving a
lower bound for the dimension of Σ which holds under very general assumptions.
5.12. Lemma. Let X n ⊂ PN be a nonsingular projective algebraic variety such
that dim X ∗ = N − 1, and let Σ = Sing X ∗ . Then either X is a quadric and Σ = ∅
or dim Σ ≥ n − 1.
Proof. Since X ∗ is a hypersurface, α ∈ X ∗ \Σ if and only if the hyperplane section
(α)∗ ∩ X has a unique nondegenerate quadratic point. If X is a hypersurface of
degree d, then the Gauss map π : X → X ∗ is finite, and it is clear that for d > 2
dim Σ = n − 1 (for d = 2 X is a quadric, X ∗ ' X, Σ = ∅).
Suppose now that n ≤ N − 2. We have already shown that in this case there
exists an irreducible subvariety SX ⊂ PX such that dim SX = dim PX − 1 = N − 2,
p(SX ) = X and π(SX ) ⊂ Σ. Thus for an arbitrary point α ∈ π(SX ) we have
dim Yα ≥ dim SX − dim π(SX ) ≥ N − dim Σ − 2. On the other hand, by the
theorem on tangencies dim Yα ≤ N − n − 1, so that N − dim Σ − 2 ≤ N − n − 1
and dim Σ ≥ n − 1. ¤
In case II one can give an upper bound for the dimension of singular locus which
is almost equal to the lower bound.
5.13. Lemma. In case II dim Σ ≤ n.
Proof. From Lemma 5.8 it follows that in case II SΣ = X ∗ . Hence from the
∗
∗
Terracini lemma it follows
¡ −1that¢ for a general point
S α ∈ X we have TX ,α ⊃
T (Σx , Σ), where x = p π (α) , T (Σx , Σ) =
TΣ,β . From Theorem 1.4 in
Chapter I it follows that
β∈Σx
dim S(Σx , Σ) = dim Σx + dim Σ + 1 = dim Σ + N − n − 1.
5. VARIETIES OF CODEGREE THREE
97
On the other hand, S(Σx , Σ) ⊂ SΣ = X ∗ so that dim Σ + N − n − 1 ≤ N − 1, i.e.
dim Σ ≤ n. ¤
5.14. Lemma. In case II there exists a nonsingular irreducible component
Σ¡0 ⊂ Σ such
¢ that for generic point x ∈ X we have Σ0 ∩ Px = Σx (and therefore
p π −1 (Σ0 ) = X, dim π −1 (Σ0 ) = N − 2, SΣ0 = X ∗ and Σ0 is nondegenerate) and
one of the following conditions holds:
II . dim Σ0 = n, N = 3n
2 + 2;
0
II . dim Σ0 = n − 1, N = 3n
2 +1
(in particular, n is always even).
¡
¢
Proof. Let Σ0 ⊂ Σ be a component for which p π −1 (Σ0 ) = X, dim π −1 (Σ0 ) =
N − 2 (Σ0 = π(SX ), where SX ⊂ PX is the subvariety considered in the proof
of Lemma 5.9). For a general point x ∈ X, Σx is a quadric, Σ0 ∩ Px ⊂ Σx and
dim (Σ0 ∩ Px ) = N − n − 2 = dim Σx , so that either Σ0 ∩ Px = Σx or Σx is a union
of two hyperplanes in Px and Σ0 ∩ Px is one of these hyperplanes. In the first case
∗
SΣ
case there exists another component Σ00 ⊂ Σ such that
¡ 0−1= X0 ,¢and in the second
−1
0
p π (Σ0 ) = X, dim π (Σ0 ) = N − 2 and (Σ0 ∪ Σ00 ) ⊃ Σx . Then Σ0 ∩ Px is the
other component of Σx , and, applying to Σ0 and Σ00 the argument given in the proof
of Lemma 5.8, we see that Σ0 and Σ00 are linear subspaces of PN ∗ . Furthermore,
S
S
S(Σ0 , Σ00 ) ⊂ X ∗ and so S(Σ0 , Σ00 ) = X ∗ because S(Σ0 , Σ00 ) ⊃ SΣx = Px = X ∗ ,
x
x
where x runs through the set of generic points of X. Hence in this case X ∗ is a
hyperplane which is clearly impossible. Thus SΣ0 = X ∗ and in particular Σ0 is a
nondegenerate variety.
From the Terracini lemma it follows that a general point of X ∗ = SΣ0 lies on an
exactly (dim Σx )-dimensional family of chords of Σ0 , and therefore (2 dim Σ0 + 1) −
dim X ∗ = dim Σx , i.e. dim Σ0 = N − n2 − 2. Hence if dim Σ0 = n, then N = 23 n + 2,
and if dim Σ0 = n − 1, then N = 32 n + 1.
It remains to verify that Σ0 is nonsingular. Let α ∈ Σ0 be an arbitrary point. We
observe that Σ0 is not a cone with vertex α since otherwise the variety X ∗ = SΣ0
would also be a cone with vertex α and X would lie in the hyperplane (α)∗ contrary
to the assumption that X is nondegenerate. Hence the cone Sα Σ0 = S(α, Σ0 ) has
dimension dim Σ0 + 1, ¡and
point β ∈ Σ0 and an arbitrary point
¡ for a¢¢general
∗
γ ∈ hα, βi \ Σ we have p p−1 (γ)
= TX ∗ ,γ = TSΣ0 ,γ . By the Terracini lemma
¡ ¡
¢¢∗
(cf. Proposition 1.10 a) in Chapter II), p π −1 (γ)
⊃ TΣ0 ,α .
0
Let Sα Σ0 be a dense open subset of general points of the cone Sα Σ0 . We put
¡
¡
¢¢
Rα = p π −1 Sα0 (Σ0 ) ⊂ X. Then
¡
¢
dim π −1 (Sα0 Σ0 ) = dim Sα0 Σ0 = dim Σ0 + 1,
n
dim Rα ≥ dim Σ0 + 1 − (N − n − 1) =
2
and, as we have just shown, hRα i∗ ⊃ TΣ0 ,α , i.e. dim TΣ0 ,α ≤ N − 1 − dim < R >∗ .
Since dim Σ0 = N − n2 − 2 ≤ dim TΣ0 ,α , from this it follows that dim hRα i ≤ n2 + 1,
where equality holds if and only if dim TΣ0 ,α = dim Σ0 , i.e. α is a nonsingular point
of Σ0 .
98
IV. SEVERI VARIETIES
We observe that Rα is not a linear subspace of PN . In fact, when β runs through
the set¡ of general
¢ points of Σ0 and γ runs through the set of general points of hα, βi,
x = p π −1 (γ) by definition runs through the set of general points of Rα , and from
the Terracini lemma it follows that (x)∗ ⊃ TΣ0 ,β . If Rα were a linear subspace
of PN , from the Bertini theorem it would follow that Σ0 ⊂ (Rα )∗ = hRα i∗ which
contradicts the nondegeneracy of Σ0 .
Thus Rα 6= hRα i and since dim Rα ≥ n2 and dim hRα i ≤ n2 + 1 we see that Rα
n
is an irreducible hypersurface in hRα i = P 2 +1 , π −1 (Sα0 Σ0 ) = PRα and α ∈ Sm Σ0 .
Since this is true for an arbitrary point α ∈ Σ0 , Σ0 is a nonsingular variety. ¤
5.15. Lemma. In case II X n is a Severi variety, and in case II0 X n is an
3n
isomorphic projection of a Severi variety X̃ n to P 2 +1 (we use the notations of
Lemma 5.14).
Proof. In case II Σ0 is a Severi variety by definition (cf. Definition 1.2). From
the description of the structure of Severi varieties given in Remark 2.5 it follows
that X = (SΣ0 )∗ is also a Severi variety. Furthermore, SX = Σ∗0 , Σ = Σ0 . This
proves Lemma 5.15 in case II.
S
It remains to consider case II0 . As in the proof of Lemma 5.9, let S = Σx ⊂ Px ,
x
where x runs through the set of general points of X. From Lemma 5.14 it follows
that for a general point x ∈ X we have Σ0 ∩ Px = Σx , so that π(S) = Σ0 . Since
dim S = n + (N − n − 2) = N − 2 and dim Σ0 = n − 1, for each point α ∈ Σ0 we
have
dim Yα ≥ (N − 2) − (n − 1) = N − n − 1
¡
¢
(here as above Yα denotes the varieties π −1 (α) and p π −1 (α) which are naturally
isomorphic to each other). But according to the theorem on tangencies dim Yα ≤
N − n − 1. Hence dim Yα = N − n − 1 = n2 for all α ∈ Σ0 .
Let α ∈ Σ0 , y ∈ Yα be general points. Then the hyperplane (y)∗ is tangent
to X ∗ along Py , and from Lemma 5.14 and the Terracini lemma it follows that
(y)∗ is tangent to Σ0 along the quadric Σ0 ∩ Py = Σy 3 α. In particular, the
linear subspace hYα i∗ ⊂ PN ∗ is tangent to Σ0 at the point α, and therefore hYα i ⊂
³¡
¢∗ ´ n2 +1
TΣ0 ,α
. If β ∈ Σy is another point for which hα, βi 6⊂ Σy , then it is clear
that
Y
∩
Y
=
y ∈ Rα and the intersection is transverse (so that in particular
β
¡ 2¢ α
Yα X = 1). For a general point α ∈ Σ0 we have Yα ⊂ Rα , where Rα is the variety
introduced in the proof of Lemma 5.14. Since dim Yα = n2 and Rα is an irreducible
³¡
¢∗ ´ n2 +1
nonlinear hypersurface in hRα i = TΣ0 ,α
, we conclude that Yα = Rα .
Let x be a general point of X. Then the hyperplane (x)∗ is tangent to Σ0 along
the subvariety Σx ⊂ Σ0 , i.e. T (Σx , Σ0 ) ⊂ (x)∗ . From Theorem 1.4 in Chapter I it
follows that dim S(Σx , Σ0 ) = ( n2 − 1) + (n − 1) + 1 = 32 n − 1 and a general point
ξ ∈ S(Σx , Σ0 ) lies on a finite number of secants joining points of Σx with points
of Σ0 . We have ξ ∈ Py = SΣy for a suitable general point y ∈ X, and by the
above the intersection Σx ∩ Σy = Px ∩ Py reduces to a single point β = β(x, y).
n
S n
−1
We set Hx = Yα2 , where α runs through the set of general points of Σx2 .
α
Then y ∈ Hx and y lies on a unique variety Yα , α ∈ Σx , viz. on Yβ . Hence
dim Hx = ( n2 − 1) + n2 = n − 1 so that Hx is a divisor on X.
5. VARIETIES OF CODEGREE THREE
99
We observe that Hx is not a hyperplane section of X. In fact, suppose that
there exists a point ξ ∈ PN ∗ such that Hx ⊂ (ξ)∗ . Then for a general pair of points
α, β ∈ Σx we have (ξ)∗ ⊃ hYα , Yβ i, and since Yα · Yβ = x, we see that
(ξ)∗ ⊃ hTYα ,x , TYβ ,x i = TX,x ,
i.e.
since (ξ)∗ ⊃ hYα i, we have ξ ∈ hYα i∗ = TΣ0 ,α so that ξ ∈
T
T ξ ∈ Px .TFurthermore,
TΣx ,α , i.e. Σx is a cone with vertex ξ. But on the other hand
TΣ0 ,α Px =
α∈Σx
α∈Σx
the ( n2 − 1)-dimensional variety Σx is a component of the variety of ‘ends’ of secants
S n2
passing through a general point of Px (we recall that by Lemma 5.12
Px =
x∈X
3n
(X ∗ ) 2 = SΣ0 ), and therefore, for a general point x ∈ X, Σx is a nonsingular
quadric. This contradiction shows that Hx cannot be hyperplane section.
We claim that the divisor Hx is linearly equivalent to a hyperplane section. Since
Hx itself is not a hyperplane section, this would imply that the variety X is not
3n
linearly normal, i.e. X is obtained by projecting a nonsingular variety X̃ n ⊂ P 2 +2
from a point outside S X̃. By definition, X̃ is a Severi variety (in this case Σ0 is a
hyperplane section of the Severi variety Sing X̃ ∗ ' X̃).
Thus to prove Lemma 5.15 in case II0 it remains to verify that Hx is linearly
equivalent to a hyperplane section. To shorten the proof we may assume that n > 4
(to extend the proof given below to the cases n = 2 and n = 4 one needs some
additional arguments). In fact, Proposition 3 from [96] shows that the case n = 2
(in a different guise) is actually considered in the main theorem of [96]. The case
n = 4 (i.e. dim Σ0 = 3) is dealt with in [24]. For n > 4 from [54] it follows that
Pic X = Z. Hence OX (Hx ) ' OX (m), where m is a natural number. We claim that
m = 1. To prove this it suffices to verify that, for a general point α ∈ Σ0 , Hx ∩ Yα
n
∗
is¯ a hyperplane section of the hypersurface Yα2 ⊂ (TΣ0 ,α ) . Consider the map
π ¯P ∩P : PΣx ∩ PYα → Σx , where PΣx = π −1 (Σx ) ⊂ PX , PYα = p−1 (Yα ) ⊂ PX .
Yα
Σx
The fiber of this map over a point β ∈ Σx is naturally isomorphic to Yα ∩ Yβ , and
if the intersection is not transverse, then hα,¯βi ⊂ Σ0 and in particular α ∈ TΣ0 ,β .
Hence for α ∈
/ (x)∗ ⊃ T (Σx , Σ0 ) the map π ¯P ∩P is an isomorphism, and the
Yα
Σx
variety PΣx ∩ PYα is naturally isomorphic to the ( n2 − 1)-dimensional quadric Σx .
By definition, p (PΣx ∩ PYα ) = Hx ∩ Yα and the fiber over a point y ∈ Hx ∩ Yα
is naturally ¯ isomorphic to the linear subspace Σx ∩ Σy = Px ∩ Py , so that the
morphism p¯P ∩PY is birational. Now it is easy to show that the morphism
Σx
α
³ ¯
p¯P
Σx ∩PYα
´ ³ ¯
◦ π¯
P
Σx ∩PYα
´−1
: Σx → Hx ∩ Yα
is an isomorphism. Since, as it was shown above, Yα is not a linear subspace, from
this it follows that Yα is a quadric and Hx ∩ Yα ' Σx is a hyperplane section of
Yα , i.e. m = 1 and Hx ∈ |OX (1)|. This completes the proof of Lemma 5.15 and
Theorem 5.2. ¤
5.16. Remark. Let X n ⊂ PN be a nondegenerate nonsingular variety which
can be isomorphically projected to PN −1 . From Corollary 2.11 in Chapter II and
Definition 1.2 it follows that if n = 2k is even, then s = dim SX > 3k + 1, and
100
IV. SEVERI VARIETIES
equality holds if and only if X is a Severi variety. Classification of Severi varieties is
given in Theorem 4.7. If n = 2k − 1 is odd, then from Corollary 2.11 in Chapter II
it follows that s > 3k. In this case it also seems interesting to classify extremal
varieties for which s = 3k. Such classification can be accomplished by using methods of the present chapter, but the corresponding paper is not yet written. Here
we only give the expected answer (cf. also Remark 2.7 in Chapter V). Since each
nondegenerate curve in PN , N > 4 is an extremal variety, we may assume that
n > 3. Furthermore, for N = s + 3 X = v2 (P3 ) ⊂ P9 is the Veronese threefold, and
for N = s + 2 there are three possibilities: X 3 is an isomorphic projection of the
Veronese threefold v2 (P3 ) to P8 , X 3 is the image of P3 with respect to the rational
map defined by the linear system of quadrics passing through a point (i.e. X is
obtained from v2 (P3 ) by projecting it from a point lying in v2 (P 3 ); in this case X
is isomorphic to the blow up of P 3 with center at a point), X 5 = P2 × P3 ⊂ P11 is
a Segre variety. Finally (and this is the only result whose proof is not written), for
N = s + 1 X is either an isomorphic projection of one of the varieties listed above
or a hyperplane section of one of the Severi varieties X 2k , k ≥ 2. The proof should
be essentially parallel to the proof of classification theorem for Severi varieties given
in the present chapter; in particular, one can use the fact that nonsingular hyperplane sections are projectivizations of orbits of highest weight vectors of irreducible
representations of algebraic groups (viz. hyperplane sections of P2 × P2 correspond
to the adjoint representation of SL3 , hyperplane sections of the Grassmann variety
G(5, 1) of lines in P5 correspond to the second fundamental representation of Sp6 ,
and hyperplane sections of the variety E 16 ⊂ P26 correspond to the nontrivial representation of lowest possible dimension of F4 ). In the proof of Lemma 5.15 in case
II0 it would be natural to refer to classification of odd-dimensional extremal varieties as we referred to classification of Severi varieties in case II. However, for lack
of suitable reference, we argued indirectly by using additional information available in our special case (in particular, in the notations of this remark, we know in
¡
¢n+1
advance that the variety (SX)∗
is nonsingular).
CHAPTER V
LINEAR SYSTEMS OF HYPERPLANE SECTIONS
ON VARIETIES OF SMALL CODIMENSION
Typeset by AMS-TEX
101
102
LINEAR SYSTEMS OF HYPERPLANE SECTIONS
1. Higher secant varieties
Let X n ⊂ PN be a nondegenerate projective variety over an algebraically closed
field K. Put
¯
¡ k ¢0
SX = {x0 , . . . , xk ; u) ∈ X × . . . × X ×PN ¯ dim hx0 , . . . , xk i = k, u ∈ hx0 , . . . , xk i},
|
{z
}
k+1
¡ k ¢0
k
and let SX
be the closure of SX
in X × · · · × X ×PN . We denote by ϕk (or,
|
{z
}
k+1
k
to PN and by pki (or
if there is no ambiguity, simply by ϕ) the projection of SX
k
simply by pi ) the projection of SX onto the i-th factor of X × · · · × X (i = 0, . . . , k).
k
) is called the variety of k-secants
1.1. Definition. The variety S k X = ϕ(SX
of the variety X.
Thus S k X is the closure of the set of points lying in k-dimensional linear subspaces spanned by general collections of k + 1 points in X. In particular, S 0 X = X
and S 1 X = SX is the usual secant variety (cf. §1 of Chapter I).
1.2. It is clear that all S k X, 0 ≤ k ≤ k0 are irreducible projective varieties and
X ⊂ SX ⊂ S 2 X ⊂ · · · ⊂ S k X ⊂ · · · ⊂ S k0 −1 X ⊂ S k0 X = PN ,
where
¯
k0 = min {k ¯ S k X = PN }.
(1.2.1)
(1.2.2)
k
The variety S X can also be constructed inductively as follows. Let a0 ≤ · · · ≤
ar be a collection of nonnegative integers such that a0 + · · · + ar = k − r, let
SS0 a0 X,...,S ar X =
¯
©
ª
(v0 , . . . , vr ; u) ∈ S a0 X ×. . .×S ar X ×PN ¯ dim hv0 , . . . , vr i = r, u ∈ hv0 , . . . , vr i ,
and let SS a0 X,...,S ar X be the closure of SS0 a0 X,...,S ar X in S a0 X × · · · × S ar X ×
PN ⊂ PN × · · · × PN . In this case we also denote by ϕ (or, to avoid ambiguity, by
|
{z
}
r+2
ϕa0 ,...,ar ) the projection map from SS a0 X,...,S ar X to PN and by pi (or pia0 ,...,ar ) the
projection map from SS a0 X,...,S ar X to S ai X. It is not hard to see that
S k X = ϕa0 ,...,ar (SS a0 X,...,S ar X ) .
(1.2.3)
Definition 1.1 is a special case of (1.2.3) for r = k, a0 = · · · = ar = 0, so that
k
SX
= SX, . . . , X .
| {z }
k+1
We shall often use another special case of (1.2.3), viz. r = 1. In this case from
(1.2.3) it follows that, in the notations of § 1 of Chapter I,
S k X = S (S a0 X, S a1 X) ,
a0 + a1 = k − 1.
(1.2.4)
In particular, for a0 = 0 we obtain the following inductive formula:
S k X = S(X, S k−1 X).
(1.2.5)
1. HIGHER SECANT VARIETIES
103
1.3. Proposition. All inclusions in (1.2.1) are strict. In other words, S k−1 X 6=
S X for 1 ≤ k ≤ k0 .
k
Proof. Suppose that S k−1 X = S k X. Then from (1.2.5) it follows that for each
point x ∈ X the variety S k−1 X is a cone with vertex x. But this is possible only if
S k−1 X = PN contrary to (1.2.2). ¤
1.4. Proposition. Let v0 ∈ S a0 X, . . . , vr ∈ S ar X, dim hv0 , . . . , vr i = r, u ∈
hv0 , . . . , vr i ⊂ S k X, where k = a0 + · · · + ar + r, and let Lu = TS k X,u . Then
a) Lu ⊃ hTS a0 X,v0 , . . . , TS ar X,vr i;
b) if char K = 0, v0 , . . . , vr is a generic collection of points of X, and u is a
generic point of the linear subspace hv0 , . . . , vr i, then
Lu = hTS a0 X,v0 , . . . , TS ar X,vr i.
Proof. The proof is by induction on r. We use the representation
¡
¢
S k X = ϕ SS a0 X,S a1 +···+ar +r−1 X
(cf. (1.2.4)). Then
u ∈ hv0 , vi,
v ∈ hv1 , . . . , vr i,
and it suffices to apply Proposition 1.10 from Chapter II to the subvariety
S a0 X ⊂ S a1 +···+ar +r−1 X
and the points u, v0 , and v. ¤
1.5. From now on we shall assume that the variety X is nonsingular (as in
Chapter II, similar results can be proved for singular varieties, but we will not need
them). Let u ∈ S k X. We put
¡
¢
Yu = pk0 (ϕk )−1 (u) .
(1.5.1)
From Proposition 1.4 a) it follows that the linear subspace Lu = TS k X,u is tangent
to X along the subvariety Yu ⊂ X. The dimension of Yu for a generic point u ∈ S k X
is a projective invariant of the variety X; we put
δk = dim Yu .
(1.5.2)
It is easy to compute this invariant in terms of the dimensions of higher secant
varieties. To wit, let
sk = dim S k X
k
(in¡ particular,
¢ s0 = n, s1 = s = dim SX). We use the representation S X =
ϕ SX,S k−1 X (cf. (1.2.5)). Then
¡
¢
Yu = p00,k−1 (ϕ0,k−1 )−1 (u) ,
and if 1 ≤ k ≤ k0 , (so that by Proposition 1.3 S k−1 X 6= S k X), then
¡
¢
δk = dim Yu = dim (ϕ0,k−1 )−1 (u)
= dim SX,S k−1 X − dim S k X = sk−1 + n + 1 − sk
(1.5.4)
104
LINEAR SYSTEMS OF HYPERPLANE SECTIONS
(in particular δ1 = 2n + 1 − s; for the sake of brevity we shall denote δ1 simply by
δ).
From Proposition 1.4 a) it follows that for k < k0
δk < n;
(1.5.5)
δk0 ≤ n,
(1.5.6)
it is clear that
and in view of (1.5.4) equality in (1.5.6) holds if and only if
sk0 −1 = N − 1;
(1.5.7)
finally, sk = N for k ≥ k0 , and δk = n for k > k0 .
Summing up the equalities (1.5.4) for all 1 ≤ k 0 ≤ k, we see that for k ≤ k0
sk = (k + 1)(n + 1) −
k
X
i=1
δi − 1 =
k
X
(n − δi + 1) − 1
(1.5.8)
i=0
(we recall (cf. (1.5.1), (1.5.2)) that
δ0 = dim Yx = dim x = 0,
(1.5.9)
where x ∈ X is a generic point).
1.6. Example. Let n = 1, so that X is a curve. From (1.5.5) it follows that
δk = 0 for k < k0 . Formula (1.5.8) shows that
sk = min {2k + 1, N }.
Thus the dimensions of higher secant varieties of a curve do not depend on its
properties. Below we shall see that for varieties of higher dimensions this is far
from being so.
1.7. Proposition. 0 = δ0 ≤ δ1 = δ ≤ δ2 ≤ · · · ≤ δk0 −1 ≤ δk0 ≤ n, δk0 −1 ≤
n − δ, and δk = n for k > k0 .
Proof. Let k ≤ k0 . Consider the rational maps
ξ : SX,X,S k−2 X 99K SX,S k−2 X ,
ξ(x0 , x, w, u) = (x0 , w, hx0 , wi ∩ hx, ui)
η : SX,X,S k−2 X 99K SX,S k−1 X ,
η(x0 , x, w, u) = (x, hx0 , wi ∩ hx, ui, u).
and
It is clear that ξ and η are defined off (ϕ0,0,k−2 )−1 (X). Furthermore,
0,0,k−2
p0,k−2
◦ξ = p2
,
1
ϕ0,k−1 ◦η = ϕ0,0,k−2 .
1. HIGHER SECANT VARIETIES
105
Let x be a general point of X, let v be a general point of S k−1 X, and let u ∈ hx, vi
be a general point of S k X. Then
¡
¢
(ϕ0,0,k−2 )−1 (u) = η −1 (ϕ0,k−1 )−1 (u) ⊃ η −1 (x, v, u),
¡
¢ ¡
¢−1
ξ η −1 (x, v, u) = ϕ0,k−2
(v),
¢ 0,0,k−2¡ −1
¢
0,0,k−2¡ 0,0,k−2 −1
Yu =p0
(ϕ
) (u) ⊃ p0
η (x, v, u)
¢¢
¡ 0,k−2 −1 ¢
0,k−2 ¡ ¡ −1
= p0
ξ η (x, v, u) =p0,k−2
(ϕ
) (v) =Yv .
0
Hence
δk = dim Yu ≥ dim Yv = δk−1 ,
i.e. the numbers δk form an increasing sequence.
It remains to show that if S k X 6= PN , then δk ≤ n − δ. Let u be a generic
point of S k X, and let Lu = TS k X,u . From Proposition 1.4 a) it follows that in the
notations of Section 1 of Chapter I T (Yu , X) ⊂ Lu . Since X is a nondegenerate
variety,
T (Yu , X) 6= S(Yu , X) ⊃ X.
Hence Theorem 1.4 from Chapter I yields:
s = dim SX ≥ dim S(Yu , X) = dim Yu + dim X + 1 = δk + n + 1,
i.e.
δk ≤ s − n − 1 = n − δ
as required.
(1.7.1)
¤
The following theorem improves the monotonicity result of Proposition 1.7.
1.8. Theorem. For 1 ≤ k ≤ k0 we have δk ≥ δk−1 + δ.
Proof. Consider the commutative diagram of rational maps
(1.8.1)
where for generic points x ∈ X, vk−2 ∈ S k−2 X, v0 ∈ X, u ∈ hx, vk−2 , v0 i ⊂ S k X
the map λ is defined by the formula
λ(x, vk−2 , v0 , u) = (vk−1 , v0 , u),
where
vk−1 = hx, vk−2 i ∩ hv0 , ui
and µ is defined by the formula
µ(x, vk−2 , v0 , u) = (vk−2 , v1 , u),
106
LINEAR SYSTEMS OF HYPERPLANE SECTIONS
Fig. 1
where
v1 = hx, v0 i ∩ hvk−2 , ui
(cf. fig. 1).
It is clear that
¡ ¡
¢¢
ϕk−1,0 λ µ−1 (vk−2 , v1 , u) = u.
Hence
¡ ¡
¢¢ ¡
¢−1
λ−1 λ µ−1 (vk−2 , v1 , u) ⊂ ϕ0,k−2,0
(u),
¢¢
¢
0,k−2,0 ¡ 0,k−2,0 −1
0,k−2,0 ¡ −1 ¡
λ
λ(vk−2 , v1 , u) ⊂ p0
(ϕ
) (u) = Yu
p0
and
¡ −1 ¡
¢¢
λ
λ(µ−1 (vk−2 , v1 , u)) .
δk = dim Yu ≥ dim p0,k−2,0
0
(1.8.2)
It is clear that
¡
¢
¡
¢
dim λ µ−1 (vk−2 , v1 , u) = dim µ−1 (vk−2 , v1 , u) = dim (ϕ0,0 )−1 (v1 ) = δ1 . (1.8.3)
On the other hand, for a generic point (vk−1 , v0 , u) ∈ SS k−1 X,X we have
¡
¢
dim λ−1 (vk−1 , v0 , u) = dim (ϕ0,k−2 )−1(vk−1 ) = δk−1 .
(1.8.4)
From (1.8.3) and (1.8.4) it follows that
¡ ¡
¢¢
dim λ−1 λ µ−1 (vk−2 , v1 , u) ≥ δk−1 + δ1 .
(1.8.5)
From (1.8.2) and (1.8.5) it is clear that in order to prove Theorem 1.8 it suffices to verify that the map p0,k−2,0
is finite at a generic point of the variety
0
¡ ¡
¢¢
¡ −1 ¡ ¡ −1
¢¢¢
λ−1 λ µ−1 (vk−2 , v1 , u) . Let y ¡∈ ¡p0,k−2,0
λ
λ µ
(vk−2 , v1 , u)
be a generic
0
¢¢
point. The preimage of y in λ−1 λ µ−1 (vk−2 , v1 , u) consists of quadruples of the
0
form (y, vk−2
, v0 , u), where
0
hy, vk−2
i 3 vk−1 ,
vk−1 = hx, vk−2 i ∩ hv0 , ui,
hx, v0 i 3 v1
1. HIGHER SECANT VARIETIES
107
Fig. 2
(cf. fig. 2).
Hence it remains to show that the subvarieties
¡ k−2,0 −1
¢
Yvk−1 = pk−2,0
(ϕ
) (vk−1 ) ,
1
and
u ∈ hy, vk−1 i
¡ 0,0 −1 ¢
(ϕ )
Yv1 = p0,0
1
intersect in a finite number of points. This immediately follows from the general observation that if vk−1 is a generic point of S k−1 X, then dim S(Yvk−1 , X) =
dim Yvk−1 + n + 1. To prove this equality we argue as in the proof of the last assertion of Proposition 1.7. From Proposition 1.4 a) it follows that T (Yvk−1 , X) ⊂
TS k−1 X,vk−1 6= PN is a proper linear subspace of PN . Since X is nondegenerate,
we have S(Yvk−1 , X) 6= T (Yvk−1 , X), and our claim follows from Theorem 1.4 in
Chapter I. ¤
1.9. Corollary. For 0 ≤ k ≤ k0 we have δk ≥ kδ.
1.10. Remark. In the case when char K = 0 Theorem 1.8 can be proved using
Proposition 1.4 b) for k = r, a0 = · · · = ar = 0 and general position arguments. A
proof of this type was independently discovered by A. Holme and J. Roberts (cf.
§ 5 in [40]).
1.11 Remark. It would be interesting to find out under what conditions Theorem 1.8 can be generalized as announced in [100]: The function δ is superadditive on
the interval [0, k0 ], i.e. if k = k1 + · · · + kr is a partition of k, then δk ≥ δk1 + . . . δkr .
There is a gap in the proof of this statement given in [100], and Ådlansvik [103] observed that counterexamples are given by Dale’s surfaces [14] (cf. also [40]). Some
approaches to this problem are discussed in [40] and also in [22], where it is shown
that superadditivity holds under a rather restrictive assumption which is apparently hard to verify, viz. that the higher secant varieties of X be almost smooth,
i.e. for all z ∈ S k X TS0 k X,z ⊂ S(z, S k X). However the problem still remains open.
It seems plausible that superadditivity holds for varieties with δ > 0 (for Dale’s
surfaces δ = 0). Moreover, it might be that quite generally Dale’s surfaces yield
the only possible exceptions to superadditivity.
108
LINEAR SYSTEMS OF HYPERPLANE SECTIONS
1.12. Theorem. k0 ≤
theorem is empty).
£n¤
[ nδ ] X = PN (for δ = 0 the assertion of the
δ , i.e. S
Proof. From the definition of k0 (cf. (1.2.2)) it follows that for a ≥ k0 we have
S a X = PN , so that to prove Theorem 1.12 it suffices to verify that nδ ≥ k0 . From
Theorem 1.8 it follows that
δk0 ≥ k0 δ,
(1.12.1)
and (1.5.6) yields
δk0 ≤ n.
(1.12.2)
Combining (1.12.1) and (1.12.2), we see that k0 δ ≤ n (one could also use Theorem 1.8 for k = k0 − 1 and the inequality (1.7.1) proved in Proposition 1.7). ¤
Theorem 1.12 allows to give a simple proof of the Hartshorne conjecture on linear
normality (cf. Corollary 2.11 in Chapter II).
1.13. Corollary. If SX 6= PN , then δ ≤ n2 and n ≤ 23 (N − 2).
£ ¤
Proof. In fact, for δ > n2 we have nδ = 1, and from Theorem 1.12 it follows
that SX = PN . Furthermore, if SX 6= PN , then
N ≥ s + 1 = 2n + 2 − δ ≥
i.e. n ≤ 23 (N − 2). ¤
3n
+ 2,
2
2. MAXIMAL EMBEDDINGS OF VARIETIES OF SMALL CODIMENSION
109
2. Maximal embeddings of varieties of small codimension
Let X be a nonsingular projective variety over an algebraically closed field K.
Projections yield a partial order on the set of all nondegenerate embeddings of X
in projective spaces. Of special interest are maximal and minimal embeddings with
respect to this order.
We denote by M (n, δ) (resp. m(n, δ)) the maximal (resp. minimal) integer N for
which there exists a nonsingular nondegenerate variety X ⊂ PN such that
dim X = n,
δ(X) = δ
(in case there is no finite maximum, we set M (n, δ) = ∞). Ruling out the obvious
case X = Pn , we see that the functions m and M are defined on the set of all pairs
(n, δ) ∈ Z2 for which 0 ≤ δ ≤ n and M (n, 0) = ∞.
2.1. Example. Suppose that δ >
n
2.
Then
M (n, δ) = m(n, δ) = 2n + 1 − δ.
In fact, by Corollary 1.13 (or Corollary 2.11 from Chapter II), for δ >
n
2
SX = PN = hXi
and
N = s = 2n + 1 − δ = M (n, δ) = m(n, δ).
2.2. Proposition.
(i) m(n, δ) = 2n + 1 − δ;
(ii) M (n, δ − 1) ≥ M (n, δ) + 1;
(iii) M (n − 1, δ − 1) ≥ M (n, δ) − 1
(we assume that all pairs of integers involved in the statement of the proposition lie
in the domain of definition of the functions m and M ).
Proof. (i) In fact, for each X with dim X = n, δ(X) = δ we have
m(n, δ) ≥ sX = 2n + 1 − δ.
(2.2.1)
Taking X to be the intersection of n + 1 − δ generic hypersurfaces
Hi ⊂ P2n+1−δ ,
deg Hi > 1,
i = 1, . . . , n + 1 − δ,
we see that
m(n, δ) ≤ 2n + 1 − δ.
(2.2.2)
Combining (2.2.1) and (2.2.2) we see that (i) holds.
(ii) Let X ⊂ PM (n,δ) be a nonsingular nondegenerate variety for which dim X =
n, δ(X) = δ, and let u ∈ PM (n,δ)+1 be a generic point. Consider the projective
cone S(u, X) ⊂ PM (n,δ)+1 with vertex u, and let
X 0 = S(u, X)n+1 ∩ H M (n,δ) ,
110
LINEAR SYSTEMS OF HYPERPLANE SECTIONS
where
H ⊂ PM (n,δ)+1 ,
deg H > 1
0
is a general hypersurface. Then X is a nonsingular nondegenerate variety, n0 =
dim X 0 = dim X = n, and it is easy to see that for δ > 0
SX 0 = S(u, SX),
and
s0 = dim SX 0 = dim SX + 1 = s + 1,
δ 0 = δ(X 0 ) = 2n0 + 1 − s0 = 2n − s = δ − 1,
which proves (ii).
(iii) Let X ⊂ PM (n,δ) be a nonsingular nondegenerate variety for which dim X =
n, δ(X) = δ, and let
X 0 = X ∩ H ⊂ PM (n,δ)−1
be the intersection of X with a generic hyperplane H ⊂ PM (n,δ) . Then X 0 is a
nonsingular nondegenerate variety, n0 = dim X 0 = dim X − 1 = n − 1, and it is not
hard to see that for δ > 0
SX 0 = SX ∩ H,
and
s0 = dim SX 0 = dim SX − 1 = s − 1
δ 0 = δ(X 0 ) = 2n0 + 1 − s0 = 2n − s = δ − 1.
Assertion (iii) and Proposition 2.2 are proved. ¤
¡£ ¤¢
© ª¡
© ª
¢
2.3. Theorem. M (n, δ) ≤ f nδ
= n(n+δ+2)
+ 12 nδ δ − δ nδ − 2 =
2δ
© ª
n(n+δ+2)+²(δ−²−2)
, where f (k) = (k + 1)(n + 1) − k(k+1)
δ nδ = n
2δ£ ¤
2 © δª− 1, ² =
£
¤
(mod δ), nδ is the largest integer not exceeding nδ , and nδ = nδ − nδ .
Proof. Let X ⊂ PM (n,δ) be a nonsingular nondegenerate variety, dim X = n,
δ(X) = δ. From (1.5.8) and Corollary 1.9 it follows that for k ≤ k0
sk = (k + 1)(n + 1) −
k
X
δi − 1 ≤ (k + 1)(n + 1) −
i=1
k
X
i δ − 1 = f (k).
(2.3.1)
i=1
The graph of the function f (k) is a parabola (cf. fig. 3); f attains maximal value
2
(equal to (2n+δ+2)
− 1) for k = 2n−δ+2
= a, and f is a monotone increasing
8δ
2δ
function for 0 ≤ k ≤ a.
Fig. 3
2. MAXIMAL EMBEDDINGS OF VARIETIES OF SMALL CODIMENSION
111
From (1.2.2) and (2.3.1) it follows that
M (n, δ) = sk0 ≤ f (k0 ),
and from Theorem 1.12 it follows that k0 ≤
k0 ≤
and
hni
δ
−1≤
(2.3.2)
£n¤
£n¤
δ . If k0 < δ , then
n
−1<a
δ
´
³n´
³h n i´
−1 =f
−1<f
(2.3.3)
δ
δ
δ
¡£ ¤¢
(we observe that from (2.3.1) it follows that f nδ is an integer). From (2.3.2)
and (2.3.3) it follows that we always have
f (k0 ) ≤ f
³n
M (n, δ) ≤ f
³h n i´
δ
,
and to prove Theorem 2.3 it remains to explicitly compute f
¡£ n ¤¢
.
δ
¤
2.4. Remark. Proposition 2.2 (i) and Theorem 2.3 show that for δ(X) = δ > 0
the dimension N of the linear span of X n (n ≥ δ − 1) lies in the limits depicted in
fig. 4.
Fig. 4
112
LINEAR SYSTEMS OF HYPERPLANE SECTIONS
2.5. Remark. If δ >
shows that
n
2,
then
£n¤
δ
= 1, f
¡£ n ¤¢
δ
= 2n + 1 − δ, and Theorem 2.3
M (n, δ) = m(n, δ) = 2n + 1 − δ ≤
3n + 1
2
(cf. also Corollary 1.13).
2.6. Remark. For δ =
n
2
¡
¢
n ≡ 0 (mod 2) Theorem 2.3 yields:
N ≤f
³h n i´
δ
= f (2) =
3n
+ 2.
2
Here there are two possibilities:
¡
¢
(i) SX = PN , N = s = m n, n2¡ = 3n
¢2 + 1;
(ii) SX 6= PN , N = s + 1 = M n, n2 = f (2) =
3n
2
+ 2.
Nonsingular nondegenerate varieties X n ⊂ PN for which N > s = 3n
2 + 1 are called
Severi varieties (cf.
Definition
1.2
in
Chapter
IV).
Thus
(ii)
means
that
each Severi
¡
¢
variety lies in a 3n
2 + 2 -dimensional projective space. Of course, this also follows
from classification of Severi varieties (cf. Theorem 4.7 in Chapter IV).
2.7. Remark. From Remark 2.5 it follows that if
SX n 6= PN ,
then
δ≤
n−1
,
2
n ≡ 1 (mod 2),
s = 2n + 1 − δ ≥
3n + 3
.
2
Consider the extremal case when δ = n−1
2 . By Theorem 2.3, for n = 3 (δ = 1) we
have
n(n + 3)
N≤
= 9 = s + 3,
2
and for n > 3
N≤
3n(n + 1) 1
2
+ ·
2(n − 1)
2 n−1
Thus if n ≡ 1 (mod 2), δ =
(i)
(ii)
(iii)
(iv)
µ
¶
n−1 n−1
2
3n + 7
−
·
−2 =
= s + 2.
2
2
n−1
2
n−1
2 ,
then there are the following possibilities:
¢ 3n+3
= 2 ;
SX = P , N = s = m n, n−1
2
;
N = s + 1 = 3n+5
2
¡
¡
¢
¢
N = s + 2 = 3n+7
= M n, n−1
if n > 3 ;
2
2
n = 3, N = s + 3 = M (3, 1) = 9.
N
¡
All these cases really occur: as an example of case (ii) one can take a nonsingular
hyperplane section of any of the Severi varieties; an example of case (iii) is given
by the five-dimensional Segre variety P2 × P3 ⊂ P11 , and an example of case (iv)
is given by the Veronese variety v2 (P3 ⊂ P9 (cf. [77] and [24], where all threefolds
X 3 ⊂ PN with δ = 1 and SX 6= PN are classified; cf. also Remark 2.9).
2. MAXIMAL EMBEDDINGS OF VARIETIES OF SMALL CODIMENSION
113
2.8. Definition. Nonsingular nondegenerate varieties X n ⊂ PN for which
δ(X) = δ > 0 and N = M (n, δ) will be called extremal varieties.
2.9. Remark. The bound in Theorem 2.3 is sharp. Examples of extremal
varieties X n ⊂ PN for which
³h n i´
N = M (n, δ) = f
(2.9.1)
δ
are given by all varieties with δ > n2 (cf. Remark 2.5), by the Severi varieties (δ =
cf. Remark 2.6 and Theorem 4.7 in Chapter IV), by the Veronese varieties
v2 (Pn ) ⊂ P
n(n+3)
2
n
2;
(δ = 1),
by the Segre varieties
Pa × Pb ⊂ P(a+1)(b+1)−1 ,
|a − b| ≤ 1
(δ = 2),
and by the Grassmann varieties
G(m, 1) ⊂ P (
m+1
2
)−1 ,
(δ = 4).
In Chapter VI we shall show that each variety X n ⊂ PN satisfying condition (2.9.1)
(i.e. such that the inequality in Theorem 2.3 turns into equality) coincides with one
of the varieties listed in this remark (cf. Theorem 5.6 in Chapter VI). We remark
that from the proof of Theorem¡£
2.3¤¢it follows that for an£extremal
variety X n ⊂
¤
n
n
M (n,δ)
P
, δ = δ(X) M (n, δ) = f δ if and only if k0 = δ and the inequalities
£ ¤
(2.3.1) turn into equalities for all k ≤ k0 , i.e. δi = iδ for 0 ≤ i ≤ k0 = nδ (cf.
Proposition 1.2 in Chapter VI).
2.10. Theorem. Let X n ⊂ Pr be a (not necessarily nondegenerate) nonsingular
variety. Then the dimension
of the
³h
i´ complete linear system of hyperplane sections
n
of X does not exceed f 2n+1−r , where f is the function defined by (2.3.1). In
other words,
µ·
0
h (X, OX (1)) ≤ f
n
2n + 1 − r
¸¶
·
¸
(4n − r + 3)2
+1≤
.
8(2n − r + 1)
Proof. Let X̃ ⊂ PN be the image of X under the embedding defined by the
complete linear system |OX (1)|, N = h0 (X, OX (1)) − 1. Then
δ̃ = δ(X) = 2ñ + 1 − s̃ = 2n + 1 − s ≥ 2n + 1 − r
ñ = dim X̃ = n,
s̃ = dim S X̃ = dim SX = s.
By Proposition 2.2 (ii),
N ≤ M (n, δ̃) ≤ M (n, 2n + 1 − r).
(2.10.1)
114
LINEAR SYSTEMS OF HYPERPLANE SECTIONS
From (2.10.1) and Theorem 2.3 it follows that
µ·
¸¶
n
0
h (X, OX (1)) ≤ f
+ 1.
2n + 1 − r
(2.10.2)
On the other hand, it is not hard to see that, for fixed n and r, the second term in
the expression
µ·
¸¶
n
n(3n − r + 3) ²(2n − r − ² − 1)
=
f
+
2n + 1 − r
2(2n − r + 1)
2(2n − r + 1)
attains maximal value equal to
µ·
(2n−r−1)2
8(2n−r+1)
for ² =
2n−r−1
.
2
Hence
¸¶
n(3n − r + 3) (2n − r − 1)2
(4n − r + 3)2
+
+1=
.
2(2n − r + 1)
8(2n − r + 1)
8(2n − r + 1)
(2.10.3)
Theorem 2.10 immediately follows from (2.10.2) and (2.10.3). ¤
f
n
2n + 1 − r
+1≤
2.11. Definition. Let X n ⊂ Pr be a nonsingular nondegenerate variety. The
variety X̃ n ⊂ PN , N = h0 (X, OX (1)) − 1 defined as the image of X under the
embedding given by the complete linear system |OX (1)| will be called the linear
normalization of X .
n
r
2.12. Corollary.
¡n+2¢ Let X ⊂ P , r ≤ 2n be a nonsingular variety. Then
h (X, OX (1)) ≤ 2 . In other words, if codimPr X ≤ dim X and X̃ is the linear
.
span of X, then dim hX̃i ≤ n(n+3)
2
0
Proof. Since in the statement of Theorem 2.10 X is not supposed to be nondegenerate, we may assume that r = 2n. In this case Corollary 2.12 immediately
follows from Theorem 2.10 because
µ
¶
n(n + 3)
n+2
f (n) =
=
− 1.
2
2
¤
2.13. Remark. In the case when X is a complex manifold and
r = s(X) = dim SX = dim T X = 2n
Corollary 2.12 was proved in [28, 6.16] by completely different (analytic) methods.
2.14.
Remark. If for a nonsingular variety X n ⊂ P2n we have h0 (X, OX (1)) =
¡n+2
¢
n
n
2n
is defined by a generic collection
2 , then X ' P and the embedding P ,→ P
of 2n + 1 quadratic forms (cf. Remark 2.9 and Corollary 2.9 in Chapter VI).
2.15. Remark. It is clear that for r > 2n the dimension h0 (X, OX (1)) can
assume arbitrarily large values: it suffices to take a nonsingular linearly normal
variety X̃ ⊂ PN , N ≥ r (dim X̃ = n, h0 (X̃, OX̃ (1)) = N + 1) and to project it
isomorphically to Pr ; the image of X̃ under this projection is a nonsingular variety
X ⊂ Pr for which dim X = n and h0 (X, OX (1)) = N +1. This is only a restatement
of the fact that M (n, 0) = ∞.
It is convenient to interpret Theorem 2.10 using the notion of abnormality index.
2. MAXIMAL EMBEDDINGS OF VARIETIES OF SMALL CODIMENSION
115
2.16. Definition. Let X n ⊂ Pr be a nonsingular nondegenerate variety, and
let X̃ ⊂ PN be the linear normalization of X. The number λ(X) = N − r =
h0 (X, OX (1)) − r − 1 is called the (linear) abnormality index of the projective
variety X.
It is clear that linearly normal varieties have abnormality index zero; if δ > 0,
then curves have abnormality index zero, the abnormality index of surfaces does
not exceed one, and the abnormality index of threefolds lies in the interval between
zero and three (cf. Remarks 2.5–2.7).
Theorem 2.10 can be restated as follows.
2.17. Corollary. For a nonsingular nondegenerate variety X n with s(X) ≤ s
the following inequality holds:
(2s − 3n)(s − n − 1) ²(2n − s − ² − 1)
+
=
2(2n − s + 1)
2(2n − s + 1)
µ
¶½
¾ µ
¶½
¾2
(2s − 3n)(s − n − 1)
s+1
n
s−1
n
+ n−
− n−
,
2(2n − s + 1)
2
2n − s + 1
2
2n − s + 1
λ(X) ≤
where ² = n (mod 2n − s + 1). In particular, if s(X) ≤ 2n (or, which is equivalent,
SX = T X cf. Theorem 1.4 in Chapter I ), then λ(X) ≤ n(n−1)
.
2
CHAPTER VI
SCORZA VARIETIES
Typeset by AMS-TEX
116
1. PROPERTIES OF SCORZA VARIETIES
117
1. Properties of Scorza varieties
1.1. Definition. Let X n ⊂ PN be a nonsingular nondegenerate variety. We
call X a Scorza variety if, in the notations of Chapter V,
(i) N > m(n, δ), where δ = δ(X) = 2n + 1 − s;
(ii) N = M (n, δ) < ∞, i.e. δ > 0 and X is an extremal variety in the sense of
Definition 2.8¡£from
¤¢ Chapter V;
(iii) M (n, δ) = f nδ = n(n+δ+2)+ε(δ−ε−2)
, where f is the function defined by
2δ
formula (2.3.1) from Chapter V (i.e. f (k) = (k + 1)(n + 1) − k(k+1)
δ − 1)
2
and ε = n (mod δ).
By Proposition 2.2 from Chapter V, condition (i) is equivalent to condition
(i0 ) SX 6= PN .
From Theorem 2.3 in Chapter V it follows that
(ii) & (iii) ⇔ (iv):
¡£ ¤¢
(iv) N = f nδ .
Finally, from the definition of f it follows that for 1 ≤ δ ≤
n
2
(iv) ⇒ (i0 )
¡£ ¤¢
(for δ > n2 we have f nδ
= f (1) = m(n, δ) = M (n, δ); cf. Remark 2.5 in
Chapter V).
¡£ ¤¢
Thus X is a Scorza variety if and only if n ≥ 2δ > 0, N = f nδ .
Scorza varieties are named in memory of the Italian mathematician Guido Scorza
who obtained pioneer results in the study of linear normalizations of varieties of
small codimension (cf. [77; 78]).
The goal of the present chapter is to give classification of Scorza varieties.
Throughout this chapter we consider varieties defined over an algebraically closed
field K, char K = 0.
1.2. Proposition. Let X n ⊂ PN be a nonsingular nondegenerate variety, and
let δ(X) = δ ≤ n2 . Consider the following conditions:
(a) X is a£ Scorza
variety;
¤
(b) k0 = nδ (where k0 =£ min
{k|S k X = PN });
¤
(c) δi = iδ for 0 ≤ i ≤ nδ (where δi is defined by formula (1.5.2) in Chapter V).
Then (a) ⇔ (b) & (c). Furthermore, (b) ⇒ (c) for n ≡ 0 (mod δ) and
(c) ⇒ (b) for n 6≡ 0 (mod δ).
Proof. The equivalence (a) ⇔ (b) & (c) immediately follows from the proof of
Theorem 2.3 in Chapter V (cf. Remark 2.9 in Chapter V).
Suppose that n ≡ 0 (mod δ) and that condition (b) holds. By Theorem 1.8 in
Chapter V,
iδ ≤ δi ≤ δk0 − (k0 − i)δ ≤ n − (k0 − i)δ = iδ
0≤i≤
n
,
δ
118
VI. SCORZA VARIETIES
i.e. δi = iδ for 0 ≤ i ≤ nδ . Hence for n ≡ 0 (mod δ) we have (b) ⇒ (c) and
(a) ⇔ (b).
Suppose
now that n 6= £0 ¤(mod δ) and
condition (c) holds. Then δk0 +1 =
£ nthat
¤
£ ¤
k
≥
.
On
n > nδ . Hence k0 + 1 > nδ £ and
0
δ
¤
£ ¤the other hand, by Theorem 1.12
in Chapter V we have k0 ≤ nδ . Hence k0 = nδ , i.e. for n 6≡ 0 (mod δ) we have
(c) ⇒ (b) and (a) ⇔ (c). ¤
1.3. Remark. If n 6≡ 0 (mod δ), then in general (b) 6⇒ (a). An example is given
2
by the projection of Segre variety Pm × Pm+1 ⊂ Pm +3m+1 from a generic point
2
u ∈ Pm +3m+1 . We denote this variety by X. Then
n = 2m + 1,
N 2 = m2 + 3m,
δ = 2,
δi = 2i,
but
δm = 2m + 1,
N =f
³h n i´
2
0 ≤ i < m,
k0 = m,
− 1.
If n ≡ 0 (mod δ), then in general (c) 6⇒ (a). An example is given by an arbitrary
3n
variety X n ⊂ P 2 +1 . In this case
δ=
n
,
2
δ2 = n,
but Remark 2.6 from Chapter V shows that
k0 = 1,
N = f (2) − 1.
¡£ ¤¢
1.4. Theorem. Let X n ⊂ PN , N = f nδ , δ = δ(X) be a Scorza variety,£ and
¤
let u be a general point of S k X, 2 ≤ k ≤ k0 − 1 (by Proposition 1.2, k0 = nδ ).
Then in the notations of Chapter V (cf. Chapter V, (1.5.1), (1.5.2), and (2.3.1)),
Yu ⊂ Pu is a Scorza variety, where
¯
©
ª
Pu = u0 ∈ S k X ¯ TS k X,u0 = TS k X,u = S k Yu ,
and we have
dim Yu = kδ,
k0 (Yu ) = k,
δi (Yu ) = iδ,
0 ≤ i ≤ k,
dim Pu = f (k).
If k = k0 ≥ 2, then Yu is a Scorza variety of dimension k0 δ = n − ε, δ = δ(Yu ) in
the projective space Pu = hYu i = S k0 Yu , dim Pu = f (k0 ).
Proof. From Proposition 1.2 (c) and usual general position arguments it follows
that for a generic point u ∈ S k X, 0 ≤ k ≤ k0 the variety Yu is nonsingular and
has dimension kδ. Arguing by descending induction on k, we see that it suffices to
prove Theorem 1.4 in the cases k = k0 − 1 (k0 ≥ 3) and k = k0 .
First we consider the case
k = k0 − 1,
dim Yu = (k0 − 1)δ.
1. PROPERTIES OF SCORZA VARIETIES
119
¡
Since for¢k0 ≥ 3 we have (k0 − 1)δ ≥ 2δ, from Proposition 1.2 (b) ⇒ (a) for n ≡ 0
(mod δ) it follows that to show that Yu is a Scorza variety it suffices to verify that
¸
dim Yu
= k0 − 1.
k0 (Yu ) =
δ
·
δ(Yu ) = δ,
(1.4.1)
Let
Lu = TS k0 −1 X,u ,
¡
¢
dim Lu = sk0 −1 = N − 1 − n − δk0 (X) = N − ε − 1,
where ε = n − δk0 (X) = n − k0 δ (cf. formula (1.5.4) in Chapter V). Put
¯
©
ª
YLu = x ∈ X ¯ TX,x ⊂ Lu .
By Proposition 1.4 a) from Chapter V
Yu ⊂ YLu
(1.4.2)
Projecting X to Ps (s = 2n + 1 − δ) from a general (N − s − 1)-dimensional
linear subspace of Lu and applying the theorem on tangencies (cf. Corollary 1.8 in
Chapter I), we see that
dim YLu ≤ (s − ε − 1) − n − 1 = n − δ − ε = (k0 − 1)δ = dim Yu .
(1.4.3)
From (1.4.2) and (1.4.3) it follows that Yu is a component of YLu . Varying u, we
see that for a generic point u0 ∈ S k0 −1 Yu
Yu0 = Yu
(1.4.4)
(Yu0 ⊂ YLu0 = YLu in view of Proposition 1.4 in Chapter V). From (1.4.4) it follows
that for a generic collection of k points
(y0 , . . . , yk−1 ),
yi ∈ Yu ,
0 ≤ i ≤ k − 1,
k ≤ k0
and a generic point v ∈ hy0 , . . . , yk−1 i we have Yv ⊂ Yu . In particular, for k = 2
from this it follows that
δ(Yu ) ≥ δ(X) = δ.
(1.4.5)
Since u is a generic point of S k0 −1 X,
δ(Yu ) ≤ δ
(1.4.6)
Combining (1.4.5) and (1.4.6) we conclude that
δ(Yu ) = δ
which proves the first of the equalities (1.4.1).
The fact that
¯
©
ª
Pu = u0 ∈ S k0 −1 X ¯ TS k0 −1 X,u0 = Lu
(1.4.7)
120
VI. SCORZA VARIETIES
is a linear subspace immediately follows from the reflexivity theorem of C. Segre
(cf. e.g. [49; 50]). From Proposition 1.4 b) in Chapter V it follows that
Yu ⊂ S k0 −1 Yu ⊂ Pu .
On the other hand, assertion a) of the same proposition shows that for a generic
point u0 ∈ Pu
Yu0 ⊂ YLu .
(1.4.8)
Since Yu is irreducible, from (1.4.3) and (1.4.8) it follows that
S k0 −1 Yu = Pu .
(1.4.9)
It remains to show that k0 (Yu ) = k0 − 1 (this is the second of the inequalities
(1.4.1)) and dim Pu = f (k0 − 1). In view of Proposition 1.2, to do this it suffices to
verify that
S k Yu 6= Pu ,
k < k0 − 1.
(1.4.10)
But (1.4.10) immediately follows from the fact that u is a generic point of S k0 −1 X,
so that u ∈ Pu \ S k X for k < k0 . Thus Theorem 1.4 holds for k = k0 − 1 and
therefore for all 2 ≤ k ≤ k0 − 1. We observe that the same argument shows that
equalities (1.4.1) also hold for k = 1, so that for a generic point z ∈ SX the variety
Yz is nonsingular,
¯
©
ª
dim Yz = δ(Yz ) = δ,
SYz = Pz = z 0 ∈ SX ¯ TSX,z0 = TSX,z
(1.4.11)
(from (1.4.11) it immediately follows that Yz is a hypersurface).
Now we consider the case k = k0 . To prove (1.4.7) it suffices to verify that for
a generic point u ∈ S k0 X = PN , a generic pair of points x, y ∈ Yu , and a generic
point z ∈ hx, yi we have Yz ⊂ Yu . Consider the morphism
ϕ1,k0 : SSX,S k0 −2 X → PN
(cf. §1 of Chapter V) and put
¡
¢
Yu1 = p11,k0 −2 (ϕ1,k0 −2 )−1 (u) .
From 1.2 (c) and formula (1.5.4) in Chapter V it follows that
dim Yu1 = (s1 + sk0 −2 + 1) − sk0
= (2n + 2 − δ) + sk0 −2 − (sk0 −1 + n + 1 − δk0 )
= (2n + 2 − δ) − [(sk0 −2 + n + 1 − δk0 −1 ) + (n + 1 − δk0 )]
= δk0 −1 + δk0 − δ = 2(k0 − 1)δ. (1.4.12)
From (1.4.12) it follows that
dim Yu1 + 2δ = 2 dim Yu ,
and hence there exists a point z̃ ∈ Yu1 such that x, y ∈ Yz̃ ⊂ Yu . Furthermore, it is
clear that z̃ is a generic point of Pz , and from (1.4.11) it follows that Yz = Yz̃ ⊂ Yu
which implies (1.4.7). From the already established part of Theorem 1.4 it follows
that
dim Yu
k0 (Yu ) = k0 (X) = k0 =
.
δ¢
¡
Hence Proposition 1.2 (b) ⇒ (a) for n ≡ 0 (mod δ) shows that Yu is a Scorza
variety. ¤
1. PROPERTIES OF SCORZA VARIETIES
121
1.5. Corollary. Let X n ⊂ PM (n,δ) , δ = δ(X) be a Scorza variety. Then for a
generic point z ∈ SX the variety Yz is a nonsingular δ-dimensional quadric.
Proof. In the proof of Theorem 1.4 it was shown that for a generic point z ∈ SX
the variety Yz is a nonsingular hypersurface in Pz = SYz = Pδ+1 (cf. (1.4.11)).
Since X is not a hypersurface, a generic secant intersects X at exactly two points
(cf. [34, Chapter IV, §3; 39, 2.5; 64, §7 B]) which proves Corollary 1.5 (in view
of Proposition 2.1 in Chapter IV, Corollary 1.5 also follows from Corollary 1.6
below). ¤
1.6. Corollary. Let X n ⊂ PM (n,δ) , δ = δ(X) be a Scorza variety. Then δ = 1,
2, 4, or 8, and if u ∈ S 2 X is a generic point, then Yu is a Severi variety (cf.
Definition 1.2 in Chapter IV).
Proof. From Theorem 1.4 it follows that Yu is an extremal variety,
dim Yu = 2δ,
δ(Yu ) = δ,
dim hYu i = f (2) = 3δ + 2.
(1.6.1)
From Remark 2.6 in Chapter V it follows that Yu is a Severi variety. In view of
Theorem 3.10 in Chapter IV, from (1.6.1) it follows that δ = δ(Yu ) can assume only
four values indicated in the statement of the corollary. ¤
Corollary 1.6 shows that for classification of Scorza varieties it suffices to consider
four cases, viz. δ = 1, 2, 4, 8. This will be done in the next four sections.
122
VI. SCORZA VARIETIES
2. Scorza varieties with δ = 1
The goal of the present section is to give a proof of the following main result.
2.1. Theorem. Let X n ⊂ PN , n ≥ 2 be a nonsingular nondegenerate variety
such that sX ≤ 2n (by Theorem 1.4 of Chapter I, this condition holds if and only
if SX = T X). Then N ≤ n(n+3)
and equality holds if and only if X = v2 (Pn ) ⊂
2
P
n(n+3)
2
is the Veronese variety.
In particular, M (n, 1) = n(n+3)
and the Veronese variety is the only Scorza
2
variety of dimension n with δ = 1.
Proof. The inequality N ≤ n(n+3)
was already proven in Corollary 2.12 in Chap2
ter V.
Suppose that N = f (n) = n(n+3)
. From Proposition 2.2 in Chapter V it follows
2
that that δ = 1. We start with the following general result which holds for all δ.
2.2. Lemma. Let X n ⊂ PM (n,δ) , δ = δ(X) > 0 be a Scorza variety. Suppose
that n ≡ 0 (mod δ) so that M (n, δ) = f (k0 ), k0 =¡ nδ , and let¢ u ∈ S k0 −1 X and
z ∈ SX be generic points. Then sk0 −1 = N − 1 and Yzδ · Yun−δ = 1.
Proof of Lemma 2.2. From Proposition 1.2 (c) and formulas (1.5.6) and (1.5.7)
in Chapter V it follows that
n = δk0 = k0 δ,
sk0 −1 = sk0 − (n + 1 − δk0 ) = N − 1.
From (1.4.2) it follows that
T (Yu , X) ⊂ Lu = TS k0 −1 X,u .
(2.2.1)
Since X is a nondegenerate variety, from (2.2.1) and Theorem 1.4 of Chapter I it
follows that
dim S(Yu , X) = dim Yu + n + 1 = 2n + 1 − δ = s.
(2.2.2)
Equality (2.2.2) means that
S(Yu , X) = SX
k0 −2
(similarly, one can show that S(Yz , S
X) = S
that
Yz ∩ Yu 6= ∅.
Yzδ
(2.2.3)
k0 −1
X). From (2.2.3) it follows
(2.2.4)
Pδ+1
z
In view of Corollary 1.5,
⊂
= SYz is a nonsingular quadric. Since Lu ⊃
6 Pz ,
Lu ∩ Pz is a hyperplane in Pz which is tangent to Yz at all points of Yz ∩ Yu . From
this and (2.2.4) it immediately follows that (Yz · Yu ) = 1.
Lemma 2.2 is proved.
We return to the case δ = 1. In this case (2.2.1) means that Lu · X = rYu + Eu ,
where r ≥ 2, Yu 6⊂ Supp Eu . From Lemma 2.2 it follows that
2 = deg Yz = (Lu · Yz ) = r + (Eu · Yz ).
(2.2.5)
Formula (2.2.5) shows that
r = 2,
(Eu · Yz ) = 0.
(2.2.6)
2. SCORZA VARIETIES WITH δ = 1
123
2.3. Lemma. In the assumptions of Theorem 2.1, for a generic point u ∈
S n−1 X we have Eu = 0.
Proof of Lemma 2.3. First we observe that if Eu 6= 0, then Eu = E is the fixed
part of the algebraic system Lu · X, where u runs through the set of generic points
of S n−1 X. In fact, if this were not so, then Eu would contain a generic point x ∈ X,
and if x0 is another generic point of X and z is a generic point of the line hx, x0 i,
then Yz would intersect the divisor Eu at a finite set of points. Since x ∈ Yz ∩ Eu ,
this set is non-empty contrary to (2.2.6).
Lemma 2.3 now follows from the following result.
2.4. Lemma. Let X n ⊂ PM (n,δ) , δ = δ(X) > 0 be a Scorza variety. Then the
algebraic system of divisors Hu = (Lu · X)PN , Lu = TS k0 −1 X,u , u ∈ S k0 −1 X does
not have fundamental points on X.
Proof of Lemma 2.4. Suppose the converse, and let y be a fundamental point.
Then
\
y∈
TS k0 −1 X,u ,
u∈S k0 −1 X
and therefore S k0 −1 X is a cone with vertex y. Hence Lemma 2.4 is a consequence
of the following result.
2.5. Lemma. Let X n ⊂ PM (n,δ) , δ = δ(X) > 0 be a Scorza variety. Then
S
X is not a cone.
k0 −1
Proof. We prove the lemma by induction. If n = 2δ, then X is a Severi variety
and Lemma 2.5 follows from results of Chapter IV. Using Theorem 1.4, suppose
that for a generic point u ∈ S k0 −1 X the variety S k0 −2 Yu is not a cone and the
variety S k0 −1 X is a cone with vertex v. Let u0 be a generic point of the line hv, ui.
Then
u0 ∈ S k0 −1 X,
Lu0 = TS k0 −1 X,u0 = TS k0 −1 X,u = Lu
and hence Yu0 = Yu (cf. (1.4.4)). For a generic point x ∈ Yu , we consider the curve
0 −2
Cv,x,u = p0,k
1
³¡
¢−1 ¡
¢´
p0,k0 −2 × ϕ0,k0 −2
x × hv, ui0 ⊂ S k0 −2 Yu
(2.5.1)
in the plane
Π = hv, x, ui ⊂ S k0 −1 Yu ,
(2.5.2)
where hv, ui0 denotes the set of generic points of the line hv, ui. Since sk0 −2 ≤
sk0 −1 − 2 (cf. formula (1.5.4) in Chapter V), we may assume that
hv, ui ∩ S k0 −2 Yu = hv, ui ∩ S k0 −2 X = v.
(2.5.3)
From (2.5.1) and (2.5.3) it follows that hv, ui ∩ Cv,x,u = v, and since u is a generic
point of Π, Cv,x,u consists of several lines passing through v. Hence for a generic
point w ∈ Suk0 −2 we have hv, wi ⊂ S k0 −2 Yu , i.e., contrary to our assumption,
S k0 −2 Yu is a cone with vertex v.
This contradiction proves Lemma 2.5 and therefore also Lemmas 2.4 and 2.3.
124
VI. SCORZA VARIETIES
From Lemma 2.3 it follows that
Lu · X = 2Yu ,
(2.5.4)
so that in particular Yu is an ample divisor on X. Now it is easy to prove Theorem 2.1 by induction on n using Theorem 1.4, Corollary 1.6, the classical result
of Severi for n = 2 (cf. [82, no 8] or Theorem 4.7 a) in Chapter IV), and the well
known theorem to the effect that if a nonsingular variety X n , n > 2 contains an
ample divisor Y ' Pn−1 , then X ' Pn (cf. [63]). However we give a direct proof of
Theorem 2.1.
From (2.5.4) it follows that the image of the rational map
S n−1 99K Pic0 X,
u 7→ cl (Yu − Yu0 )
(where u0 is a fixed generic point of S n−1 X) is contained in the set of points of
order two on the Picard variety. Since S n−1 X is an irreducible variety, from this it
follows that for generic points u, u0 ∈ S n−1 X we have Yu ∼ Yu0 (where ∼ denotes
linear equivalence).
Consider the complete linear system of divisors H = |Yu | on the variety X. We
observe that a general divisor H ∈ H has the form
u ∈ S n−1 X.
H = Yu ,
(2.5.5)
In fact, let x0 , . . . , xn−1 ∈ H be a generic collection of n points, and let u ∈
hx0 , . . . , xn−1 i be a generic point. Then it is clear that u is a generic point of
S n−1 X, and from Proposition 1.4 b) of Chapter V it follows that
Lu = hTX,x0 , . . . , TX,xn−1 i.
(2.5.6)
On the other hand, since X is an extremal variety, X is linearly normal, and
therefore the divisor 2H ∼ 2Yu is cut by a hyperplane LH ⊂ PN , i.e. 2H = LH · X.
Hence T (H, X) ⊂ LH , so that from (2.5.6) it follows that
LH = Lu ,
2H = LH · X = Lu · X = 2Yu ,
and H = Yu as stated in (2.5.5).
From Lemma 2.4 it follows that the linear system H does not have fundamental
points and hence defines a morphism
h : X → Pdim H .
(2.5.7)
From (1.4.3) and (2.5.5) it follows that
dim H = dim (S n−1 X)∗ = dim (X × · · · × X ) − dim (Yu × · · · × Yu ) = n. (2.5.8)
|
{z
}
|
{z
}
n
n
Since in view of (2.5.4) the linear system H is ample, (2.5.7) and (2.5.8) show that
h : X → Pn is a finite covering.
2. SCORZA VARIETIES WITH δ = 1
125
The preimage of the linear system of quadrics in Pn with respect to the morphism
h is a linear subsystem of the complete linear system |2Yu | = Lu · X on X, and
since
µ
¶
¡
¢
¡ n
¢
n+2
0
0
N + 1 = h X, OX (1) = h P , OPn (2) =
2
these two linear systems coincide with each other. Hence h is an isomorphism,
h−1 = v2 and X = v2 (Pn ). ¤
2.6. Remark. For n = 1 we have k0 = 1, so that each plane curve is extremal,
i.e. M (1, 1) = m(1, 1) = 2.
2.7. Remark. As we already pointed out, Theorem 2.1 was first proved by Severi
[82, no 8] (cf. Remark 4.11 in Chapter IV) for n = 2. Ådlandsvik independently
proved a theorem related to Theorem 2.1 to the effect that if sX ≤ 2n and S n−1 X
is not a cone (the last condition is quite hard to verify), then X is a Veronese variety
(cf. [103]).
2.8. Remark. It is clear that the morphism (2.5.7) is also defined by the linear
system |Yu0 +H|, where u0 is a fixed generic point of S n−1 X, i.e. by the linear system
of hyperplane sections of X passing through the linear subspace hYu0 i. Hence h is
induced by projecting from the subspace hYu0 i, where
dim hYu0 i = N − n − 1 =
(n − 1)(n + 2)
= f (n − 1).
2
In fact, the map inverse to the Veronese mapping
v2 : Pn → v2 (Pn ) ⊂ P
n(n+3)
2
is defined by projecting from the linear subspace
hv2 (Pn−1 )i = P
(n−1)(n+2)
2
⊂P
n(n+3)
2
(cf. Chapter III, § 3).
n
2.9. Corollary. Let X
Pir be a nonsingular variety, r ≤ 2n. Then
h ⊂
3 2
¡
¢
¡
¢
(n+
)
n+2
2
h0 X, OX (1) ≤ 2 =
with equality holding if and only if r = 2n
2
and either n = 1 or X ,→ P2n is the embedding of the projective space Pn ' X
defined by a collection of 2n + 1 quadratic forms Q0 , . . . , Q2n on Pn .
126
VI. SCORZA VARIETIES
3. Scorza varieties with δ = 2
In this section we prove the following result.
3.1. Theorem. Let X n ⊂ PN , n ≥ 4 be a nonsingular nondegenerate variety.
Suppose that sX < 2n. Then
N ≤ m(m + 2) = (m + 1)2 − 1
for n = 2m ≡ 0 (mod 2),
N ≤ (m + 1)(m + 2) − 1
for n = 2m + 1 ≡ 1 (mod 2).
Furthermore, the inequalities turn into equalities if and only if X = Pm × Pm ⊂
2
Pm(m+2) or X = Pm × Pm+1 ⊂ Pm +3m+1 is a Segre variety.
n+nmod2
n−nmod2
2
In particular, M (n, 2) = n(n+4)−nmod2
×P 2
and the Segre variety P
4
is the only n-dimensional Scorza variety with δ = 2.
Proof. The inequality
N ≤f
³h n i´
2
=
n(n + 4) − n mod 2
4
splitting into the pair of inequalities given in the statement of the theorem follows
from Proposition 2.2 (ii)¡£
and¤¢Theorem 2.3 in Chapter V.
Suppose that N = f n2 . From Proposition 2.2 in Chapter V it follows that
δ = 2. Proposition 1.2 shows that
k0 =
hni
2
= m,
δk = 2k,
0 ≤ k ≤ m.
Suppose first that n = 2m. By Lemma 2.2, sm−1 = N −1, and for a generic point
u of the hypersurface S m−1 X ⊂ PN (N = m(m+2)) the hyperplane Lu = TS m−1 X,u
is tangent to X along Yu , codimX Yu = 2. Furthermore, for a generic point z ∈ SX
we have (Yu · Yz ) = 1. Put Hu = Lu · X. For a generic point y ∈ Yu and a generic
point z ∈ Sy X
(Yz · Hu )X = (Yz · Lu )PN = (Yz · TYz ,y )P3z
(here, according to Corollary 1.5, Yz is a nonsingular two-dimensional quadric),
Pz = hYz i. Thus
(Yz · Hu )X = l1z ∪ l2z ,
where l1z , l2z is a pair of lines intersecting at y. If z 0 is another generic point of Sy X
and
(Yz0 · Hu )X = (Yz · Hu )X ,
(3.1.1)
then it is clear that Yz0 = Yz , so that z 0 ∈ P3z . Varying z in the set of general points
of the cone Sy X, we obtain a (dim Sy X − dim P3z )=(n − 2)-dimensional family of
pairs of lines l1z , l2z ⊂ Hu such that
l1z ∩ Yu = l2z ∩ Yu = (l1z · l2z )Y = y.
3. SCORZA VARIETIES WITH δ = 2
127
Furthermore, we may assume that the pairs (l1z , l2z ) are ordered so that the lines l1z
(resp. l2z ), z ∈ Sy X form an irreducible family F1y (resp. F2y ) of lines in Hu passing
through y.
For generic lines l1 ∈ F1y , l2 ∈ F2y we have
l1 ∩ l2 = (Yz · Hu )X ,
where z is a generic point of the plane hl1 , l2 i. Hence the dimension of the families
F1y and F2y is equal to 12 (n − 2) = m − 1, and the closure of the subset of the divisor
Hu swept out by the lines l1z (resp. l2z ) is an irreducible m-dimensional cone C1y
(resp. C2y ) with vertex y.
Varying y in the set of generic points of Yu , we obtain two irreducible subvarieties
C1u =
\
C1y ,
y∈Yu
\
C2u =
C2y .
y∈Yu
Since a general point y ∈ Yu is contained in an (m − 1)-dimensional family of lines
from Fiy , a general point x ∈ Ciu is also
in at most (m − 1)-dimensional
T contained
subset of lines from the family Fi =
Fiy (i = 1, 2). Hence
y∈Yu
dim Ciu ≥ dim Yu + m − (m − 1) = 2m − 1 = n − 1,
and therefore Ciu is an irreducible component of the divisor Hu = Lu · X and
Ciu · Yz = liz
Thus
C1u 6= C2u ,
(i = 1, 2).
C1u ∩ C2u ⊃ Yu .
From the above considerations it follows that Hu is a connected divisor whose
components pairwise intersect with each other along cycles of codimension two in
X lying in Sing Hu and
(C1u + C2u · Yz ) = l1z + l2z = (Lu · Yz )PN = (Hu · Yz )X .
(3.1.2)
Let E, F be an arbitrary pair of irreducible components of Hu . Then E ∩ F ⊂
Sing Hu = YLu , i.e.
T (E ∩ F, X) ⊂ Lu .
(3.1.3)
On the other hand, from the nondegeneracy of X it follows that
S(E ∩ F, X) 6⊂ Lu .
(3.1.4)
In view of (3.1.3) and (3.1.4), Theorem 1.4 from Chapter I implies that
dim S(E ∩ F, X) = dim E ∩ F + n + 1 = 2n − 1 = dim SX.
Hence S(E ∩ F, X) = SX, and therefore, for a generic point z ∈ SX,
¡
¢
0 < card Yz · (E ∩ F ) < ∞
(3.1.5)
128
VI. SCORZA VARIETIES
(compare with the proof of Lemma 2.2). From (3.1.1), (3.1.2), and (3.1.5) it follows
that
Hu = C1u + C2u ,
(C1u · C2u )X = Yu .
(3.1.6)
Since s = 2n − 1, the variety X can be isomorphically projected to P2n−1 . From
Barth’s theorem [6] it follows that
h1 (X, OX ) = 21 h1 (X, C) = 0,
∗
H 1 (X, OX
) ,→ H 2 (X, Z),
and the Picard variety Pic0 X is trivial. Hence for general points u, u0 ∈ S m−1 X
0
we have Ciu ≡ Ciu (i = 1, 2).
Consider the complete linear systems Li = |Ciu | (i = 1, 2) on the variety X. We
observe that a general divisor D ∈ Li has the form D = Ciu , where u is a suitable
m−1
general point of SX
. In fact, from the linear normality of Scorza varieties it
u0
follows that for a generic point u0 ∈ S m−1 X we have D + C1−i
= L · X, where L is
a hyperplane in PN . Let (x0 , . . . , xm−1 ) be a general collection of m points of the
u0
variety D ∩ C1−i
, and let u be a generic point of the linear subspace hx0 , . . . , xm−1 i.
Then u is a generic point of S m−1 X, and from Proposition 1.4 b) of Chapter V it
follows that
Lu = hTX,x0 , . . . , TX,xm−1 i ⊂ L.
Hence
L = Lu ,
i.e.
D = Ciu ,
0
u
= (Lu · X),
D + C1−i
0
u
u
C1−i
= C1−i
,
i = 1, 2 ,
as required.
Since the dimension of the algebraic family of divisors Hu , u ∈ S m−1 X is equal
to m dim X − m dim Y = n, the dimensions di of the linear systems Li (i = 1, 2)
satisfy the relation
d1 + d2 = n = 2m.
(3.1.7)
Suppose for example that d1 ≤ d2 . Fixing a generic point u ∈ S m−1 X and varying a
generic point u0 ∈ S m−1 X, we obtain a d2 -dimensional family of (n−2)-dimensional
cycles
0
Yu00 = C1u ∩ C2u ⊂ C1u .
Furthermore, u00 ∈ S m−1 C1u , and therefore
d2 ≤ m dim C1u − m dim Yu00 = m.
(3.1.8)
From (3.1.7) and (3.1.8) it follows that
d1 = d2 = m.
(3.1.9)
From Lemma 2.4 it follows that the linear systems L1 and L2 do not have
fundamental points. In view of (3.1.9), the linear system Li defines a morphism
hi : X → P m
(i = 1, 2).
3. SCORZA VARIETIES WITH δ = 2
Put
129
h = h1 × h2 : X → Pm × Pm ,→ Pm(m+2) .
The preimage of the complete linear system of hyperplane sections of Pm × Pm ⊂
Pm(m+2) (the Segre embedding) with respect to the morphism h is a linear subsystem of the complete linear system of hyperplane sections of X, and since
¡
¢
¡
¢
N + 1 = h0 X, OX (1) = h0 Pm × Pm , OPm (1) ⊗ OPm (1) = (m + 1)2 ,
these two systems coincide with each other. Hence h is an isomorphism and
X n ⊂ PN = Pm × Pm ⊂ Pm(m+2)
is a Segre variety.
Consider now the case n = 2m + 1. Let u be a generic point of PN , N =
2
m + 3m + 1. By Theorem 1.4, Yu is a 2m-dimensional Scorza variety, m ≥ 2,
hYu i = Sum = Pu
is a linear subspace of PN of dimension m(m + 2), and
Y u = X ∩ Pu .
Consider the set of hyperplanes passing through the subspace Pu , and let L01 be
the corresponding linear system of hyperplane sections of X. Then
dim L01 = (m2 + 3m + 1) − (m2 + 2m) − 1 = m,
and for each hyperplane L ⊃ Pu
(L · X)PN = Yu + CL ,
where CL is a divisor on X. Put L1 = |CL |. From the linear normality of X it
follows that the linear system L1 on X is complete.
As in the case n = 2m, we can apply Barth’s theorem [6] to verify that Pic0 X =
0, so that for generic points u, u0 ∈ PN we have Yu0 ∼ Yu . In particular, from this
it follows that the linear system L1 does not depend on the choice of point u ∈ PN .
Lemma 2.4 shows that
\
Yu = ∅.
(3.1.10)
u∈PN
Hence the linear system L1 does not have fundamental points and defines a morphism h1 : X → Pm .
We denote by L2 the complete linear system |Yu |. In view of (3.1.10), L2 also
does not have fundamental points. We denote by h2 the morphism corresponding
to the linear system L2 .
We observe that the dimension of the system of divisors Yu , u ∈ PN is equal to
(m + 1) dim X − (m + 1) dim Yu = m + 1.
130
VI. SCORZA VARIETIES
From the linear normality of X and the relation
¡
¢
¡
¢
N + 1 = h0 X, OX (1) = h0 Pm × Pm+1 , OPm (1) ⊗ OPm+1 (1) = (m + 1)(m + 2)
it follows that a general divisor D ∈ L2 has the form D = Yu for some point u ∈ PN ,
and the morphism
h = h1 × h2 : X → Pm × Pm+1 ,→ Pm
2
+3m+1
is an isomorphism. In other words,
X ⊂ PN = Pm × Pm+1 ⊂ Pm
2
+3m+1
is a Segre variety. ¤
3.2. Remark. For δ = 2, n ≤ 3 we have k0 = 1, and each variety X n ⊂ P2n−1 is
extremal. In other words,
M (3, 2) = m(3, 2) = 5,
M (2, 2) = m(2, 2) = 3.
3.3. Remark. As we already observed in Chapter IV (cf. Remark 4.11), for
n = 4, X is a Severi variety, and in this case Theorem 3.1 was proved by Scorza
[77; 78] and Fujita and Roberts [25]. Ådlansvik independently proved a theorem
related to Theorem 3.1 to the effect that if n = 2m, sX ≤ 2n − 1, and S m−1 X is
not a cone (the last condition is quite hard to verify), then X is the Segre variety
Pm × Pm ⊂ Pm(m+2) (cf. [103]).
3.4. Remark. As it was shown in §3 of Chapter III, the Segre variety
Pk × Pl ⊂ P(k+1)(l+1)−1
is the image of the projective space Pk+l under the rational map
Pk+l 99K P(k+1)(l+1)−1
(3.4.1)
defined by the linear system of quadrics in Pk+l passing through a union of two
nonintersecting linear subspaces Pk−1 ⊂ Pk+l and Pl−1 ⊂ Pk+l . The rational map
(3.4.1) is a birational isomorphism , and the inverse rational map
Pk × Pl 99K Pk+l
is defined by projecting from the linear subspace Pkl−1 spanned by the subvariety
Pk−1 × Pl−1 ⊂ Pk × Pl .
3.5. Corollary.
Leti X n ⊂ Pr be a nonsingular variety, r ≤ 2n − 1. Then
h
¡
¢
2
(n+2)
, and equality holds if and only if r = 2n−1 and either n ≤
h0 X, OX (1) ≤
4
n
n+1
3 or X n ,→ P2n−1 is an isomorphic embedding of the Segre variety P[ 2 ] ×P[ 2 ] ' X
by means of the mapping (Q0 : · · · : Q2n−1 ) defined by a collection of 2n forms Q0 ,
. . . , Q2n−1 of bidegree (1, 1) with respect to coordinates of factors.
4. SCORZA VARIETIES WITH δ = 4
131
4. Scorza varieties with δ = 4
The goal of the present section is to give a proof of the following main result.
4.1. Theorem. Let X n ⊂ PN , n ≥ 8 be a nonsingular nondegenerate variety.
Suppose that sX < 2n − 2. Then
N≤
n(n + 6) + ε(2 − ε)
,
8
ε = n mod 4.
¡
¢
n(n+6)
Furthermore, equality holds if and only if n is even and X = G n2 + 1, 1 ⊂ P 2
is the Grassmann variety (under the Plücker embedding).
In particular,
for¢ n ≡ 0 (mod 2) we have M (n, 4) = n(n+6)
, and the Grassmann
2
¡n
variety G 2 + 1, 1 is the only n-dimensional Scorza variety with δ = 4.
¡£ ¤¢
Proof. The inequality N ≤ f n4 which is equivalent to the inequality given
in the statement of the theorem follows from Proposition 2.2 (ii) and Theorem 2.3
in Chapter V.
¡£ ¤¢
Suppose that N = f n4 . From Proposition 1.2 it follows that
k0 =
hni
4
,
δk = 4k,
0≤k≤
hni
4
.
Suppose first that n ≡ 0 (mod 4), i.e. n = 4k0 . We use the following result
generalizing Lemma 2.2.
4.2. Lemma. Let X n ⊂ PM (n,δ) , δ = δ(X) > 0 be a Scorza variety. Suppose
that n ≡ 0 (mod δ) (so that, in accordance with Proposition 1.2, n = k0 δ), let a, b
be natural numbers such that a + b = k0 , and let v ∈ S a X and w ∈ S b X be generic
points. Then (Yv · Yw )X = 1.
Proof of Lemma 4.2. First we show that Yv ∩ Yw 6= ∅. To do this it suffices to
verify that for a generic point w ∈ S b X we have
S(Yw , S a−1 X) = S a X.
(4.2.1)
We verify equality (4.2.1) by induction on a. For a = 1, (4.2.1) reduces to formula
(2.2.3) proved in Lemma 2.2. Assuming (4.2.1), we show that
S(Yw0 , S a X) = S a+1 X
for a generic point w0 ∈ S b−1 X. Let x be a generic point of X , and let w be a
generic point of hw0 , xi ⊂ S b X. Then from (4.2.1), Theorem 1.4, and Lemma 2.2
it follows that
¡
¢
S(Yw0 , S a X) = S Yw0 , S(Yw , S a−1 X)
¡
¢
= S S(Yw0 , Yw ), S a−1 X = S(SYw , S a−1 X)
(4.2.2)
¡
¢
= S Yw , S(Yw , S a−1 X) = S(Yw , S a X)
¡
¢
= S S(Yw , S a−1 X), X = S(S a X, X)
= S a+1 X
132
VI. SCORZA VARIETIES
as required.
In view of Proposition 1.2 c) and formula (1.5.4) from Chapter V
sa = sa−1 + n + 1 − aδ = sa−1 + k0 δ + 1 − aδ = sa−1 + bδ + 1 = sa−1 + dim Yw + 1,
and from (4.2.1) it follows that for generic points v ∈ S a X, w ∈ S b X the varieties
Yv and Yw intersect at finitely many points.
To prove Lemma 4.2 it remains to verify that the morphism
ϕ : SYw0 ,S a X → S a+1 X
is birational. This is again proved by induction on a, and, in view of the chain of
equalities (4.2.2), it suffices to show that in the commutative diagram of rational
maps
(4.2.3)
generic fibers of the morphism in the bottom are connected. But this is really
so since otherwise for a generic point v 0 ∈ S a+1 X the intersection (Yv0 · Yw )X
would consist of several distinct δ-dimensional quadrics of the form Yz , z ∈ SX (cf.
Corollary 1.5), where by Lemma 2.2
(Yz · Yv )Yv0 = 1
contrary to the induction assumption according to which
(Yv · (Yv0 · Yw )X )Yv0 = (Yv · Yw )X = 1.
¤
We need several results valid for an arbitrary Scorza variety X n ⊂ PM (n,δ) ,
δ = δ(X) > 0 such that n ≡ 0 (mod δ) (recall that by Lemma 2.2 under these
assumptions S k0 −1 X is a hypersurface in PN ).
4.3. Lemma. Let X n ⊂ PM (n,δ) , δ = δ(X) be a Scorza variety. Suppose that
n ≡ 0 (mod δ). Let 0 ≤ a ≤ k0 − 1, and let v be a generic point of S a X. Then
deg S k0 −1 X = k0 + 1, multv S k0 −1 X = k0 − a.
Proof. We argue by induction. Suppose that Lemma 4.3 holds for k00 < k0 . For
a = k0 − 1 the assertion of the lemma is obvious. Let 0 ≤ a ≤ k0 − 1, b = k0 − a,
and let v be a generic point of S a X and w a generic point of S b X. By Lemma 4.2,
Yv ∩ Yw = x ∈ X.
By Theorem 1.4, Yv and Yw are Scorza varieties, and from the induction assumption
it follows that
hx, vi ∩ S a−1 X = {x, v 0 },
hx, wi ∩ S b−1 X = {x, w0 },
(4.3.1)
4. SCORZA VARIETIES WITH δ = 4
133
Fig. 1.
where v 0 and w0 are generic points of the varieties S a−1 X and S b−1 X respectively.
Put
u = hv, wi ∩ hv 0 , w0 i ∈ S k0 −1 X
(4.3.2)
(cf. fig. 1).
Then u is a generic point of S k0 −1 X, and therefore
multv S k0 −1 X + multw S k0 −1 X + 1 ≤ d,
(4.3.3)
where
d = deg S k0 −1 X.
We claim that (4.3.3) is actually an equality, i.e.
multv S k0 −1 X + multw S k0 −1 X + 1 = d.
(4.3.4)
To show this it suffices to verify that
hv, wi ∩ S k0 −1 X = v ∪ w ∪ u.
Suppose that this is not so, and let
u0 ∈ (hv, wi ∩ S k0 −1 X) \ (v ∪ w ∪ u).
(4.3.5)
Set
u00 = hv 0 , w0 i ∩ hx, u0 i.
Then u00 ∈ S k0 −1 X, and from the genericity assumptions it follows that hx, u00 i 6⊂
S k0 −1 X and
multx S k0 −1 X ≤ d − 2.
(4.3.6)
134
VI. SCORZA VARIETIES
Thus to prove (4.3.4) it suffices to show that (4.3.6) is impossible, i.e.
multx S k0 −1 X = d − 1
(4.3.7)
(we recall that for all points v ∈ S k0 −1 X we have
multv S k0 −1 X < d
(4.3.8)
since by Lemma 2.5 the variety S k0 −1 X is not a cone).
We begin with proving (4.3.4) in the case a = 1. We need to verify that for a
generic point v ∈ SX
multv S k0 −1 X = d − 2.
(4.3.9)
In fact, by (4.3.1) and (4.3.2), for a generic point w ∈ S k0 −1 X we get a canonically
defined point u ∈ hv, wi ∩ S k0 −1 X (cf. fig.1). If multv S k0 −1 X < d − 2, then for
a general pair of points (v, w) ∈ SX × S k0 −1 X the line hv, wi intersects S k0 −1 X
transversely in at least two more points u and u0 (cf. (4.3.5)). However, since the
point u is chosen canonically, projecting S k0 −1 X from the point v onto PN −1 we
get a contradiction with the fact that S k0 −1 X is an irreducible hypersurface and
the monodromy permutes the points u and u0 .
Now (4.3.9) is proved for a generic point v ∈ SX. Actually from (4.3.2) and
(4.3.8) it follows that (4.3.9) holds for an arbitrary point v ∈ SX \ X. On the other
hand, for x ∈ X
d − 2 = multv S k0 −1 X ≤ multx S k0 −1 X ≤ d − 1.
Hence to prove (4.3.7) it suffices to show that
multx S k0 −1 X > multv S k0 −1 X.
Suppose that this is not so, and let x and y be generic points of X, u a generic
point of S k0 −1 X, and Π = hx, y, ui the plane spanned by x, y, and u. Then
Π ∩ S k0 −1 X = hx, yi ∪ C,
where C ⊂ Π is a curve passing through u, and from (4.3.9) it follows that deg C =
2. It is clear that for z ∈ hx, yi ∩ C
multz S k0 −1 X = d − 1,
(4.3.10)
and from (4.3.9) it follows that z ∈ X. Since x and y are generic points of X, from
(4.3.10) and our assumption it follows that z 6= x, y, i.e., contrary to the trisecant
lemma (cf. [39, 2.5; 34, Chapter IV, §3; 64, §7 B], the chord hx, yi intersects X in at
least three points). The resulting contradiction proves (4.3.7) and hence (4.3.4) (we
remark that (4.3.7) is a special case of (4.3.4) for a = 0).
Next we show that for generic points v ∈ S a X, w0 ∈ S k0 −a−1 X = S b−1 X
multv S k0 −1 X + multw0 S k0 −1 X = d.
(4.3.11)
4. SCORZA VARIETIES WITH δ = 4
135
Fig. 2.
In fact, it is clear that
d − 1 = multv S k0 −1 X + multw S k0 −1 X ≤ multv S k0 −1 X + multw0 S k0 −1 X ≤ d.
Suppose that
hv, w0 i ∩ S k0 −1 X 3 u,
u 6= v, w0 .
(4.3.12)
Then for each point x ∈ Yv ∩Yu (by Lemma 4.2, Yv ∩Yu 6= ∅, dim Yv ∩ Yu ≥ (a−1)δ)
and a generic point w ∈ hx, w0 i we have
w ∈ S b X,
hv, wi 6⊂ S k0 −1 X,
hv, wi ∩ S k0 −1 X 3 ũ = hv, wi ∩ hx, ui
(cf. fig. 2, where u ∈ hx, u0 i, v ∈ hx, v 0 i, u0 ∈ S k0 −2 X, v 0 ∈ S a−1 X).
On the other hand,
hv, wi ∩ S k0 −1 X 3 ū = hv, wi ∩ hv 0 , w0 i
(cf. (4.3.2)). From the genericity assumptions it follows that ū 6= ũ, i.e.
multv S k0 −1 X + multw S k0 −1 X ≤ d − 2
contrary to (4.3.4). Thus assumption (4.3.12) leads to a contradiction. Hence
hv, w0 i ∩ S k0 −1 X = v ∪ w
which yields (4.3.11).
From the system of equations (4.3.4) and (4.3.11) it follows that if v ∈ S a X,
0
v ∈ S a−1 X are generic points, 1 ≤ a ≤ k0 − 1, then
multv0 S k0 −1 X = multv S k0 −1 X + 1.
k0 −1
(4.3.13)
k0 −1
Since for a generic point u ∈ S
X we have multu S
X = 1, successive application of (4.3.13) shows that for a generic point v ∈ S a X
multv S k0 −1 X = k0 − a,
0 ≤ a ≤ k0 − 1.
(4.3.14)
Combining (4.3.4) and (4.3.14), we get
d = (k0 − a) + (k0 − b) + 1 = k0 + 1.
¤
(4.3.15)
136
VI. SCORZA VARIETIES
4.4. Lemma. Let X n ⊂ PM (n,δ) , δ = δ(X) > 0 be a Scorza variety. Suppose
that n ≡ 0 (mod δ), and let u be a generic point of the hypersurface S k0 −1 X.
−n−1
Then the projection πu : X 99K Pn with center at the subspace PN
= hYu i =
u
k0 −1
S
Yu is a birational isomorphism.
More
precisely,
let
H
=
X
∩
L
u
u , where
¯
Lu = TS k0 −1 X,u . Then πu ¯X\Hu is an isomorphism.
Proof. For a point x ∈ X we set Pu,x = hx, Pu i. We observe that for x ∈
/ Hu we
have Pu,x 6⊂ S k0 −1 X. In fact, if Pu,x ⊂ S k0 −1 X, then
x ∈ X ∩ T (Pu , S k0 −1 X) ⊂ Lu ∩ X = Hu .
It is clear that the hypersurface S k0 −1 X ∩ Pu,x ⊂ Pu,x contains the linear subspace
Pu and the cone S(x, S k0 −2 Yu ) as its components. From Lemma 4.3 it follows that
¡
¢
1 + deg S k0 −2 Yu = deg S k0 −1 X ∩ Pu,x ≤ deg S k0 −1 X = k0 + 1.
(4.4.1)
On the other hand, in view of Theorem 1.4, Lemma 4.3 can also be applied to the
Scorza variety Yu , dim Yu = (k0 − 1)δ, k0 (Yu ) = k0 − 1 ≥ 2 (the case k0 = 2 was
already considered in Chapter IV), so that
¡
¢
deg S k0 −2 Yu = k0 .
(4.4.2)
From (4.4.2) it follows that the inequality (4.4.1) is actually an equality, and therefore
S k0 −1 X ∩ Pu,x = Pu ∩ S(x, S k0 −2 Yu ).
(4.4.3)
Since
S(y, S k0 −2 Yu ) ⊂ S k0 −1 X ∩ Pu,x ,
for each point y ∈ (Pu,x \ Pu ) ∩ X, from (4.4.3) it follows that
X ∩ Pu,x = (X ∩ Pu ) ∪ x = Yu ∪ x,
¡
¢
i.e. πu−1 πu (x) = x, which implies our claim.
(4.4.4)
¤
We turn to a more detailed description of the hyperplane section Hu = Lu · X
(we recall that from Corollary 1.15 a) in Chapter I it follows that if δ > 1, then
Hu = Lu ∩ X is a reduced variety). From Lemma 2.2 it follows that for a generic
point z ∈ SX the intersection Yz ∩ Yu consists of a single point y. Furthermore,
(Hu · Yz )X = (Lu · Yz )PN = (TYz ,y · Yz )Pz
is a cone with vertex y over a nonsingular (δ − 2)-dimensional quadric.
S
Let y ∈ Yu be a generic point, and let Cy = (Hu · Yz )X , where z runs through
z
the set of general points of the cone S(y, X). Then Cy is a cone with vertex y whose
base is an irreducible variety By such that
2 dim By − 2(δ − 2) = n − δ
4. SCORZA VARIETIES WITH δ = 4
137
(here n − δ = [(n + 1) − (δ + 1)] is the dimension of the family of quadrics Yz passing
through y), i.e.
dim By =
n+δ−4
,
2
dim Cy =
n+δ−2
.
2
Varying y in the set of general points of Yu , we obtain an irreducible variety C u =
S
Cy ⊂ Hu . Furthermore, if x1 , x2 is a general pair of points of C u and z is a general
y
point of hx1 , x2 i, then Yz ∩ Lu is the cone over a nonsingular (δ − 2)-dimensional
quadric with vertex at the (unique) intersection point y ∈ Yz ∩ Yu . In other words,
Bxu1 ∩ Bxu2 = y, where
©
­
ª
Bxui = y ∈ Yu xi , y i ⊂ X ,
i = 1, 2.
(4.4.5)
From this it follows that
dim Yu
,
(4.4.6)
2
¡
¢
i.e. a generic point x ∈ C u is contained in a n−δ
-dimensional family of lines of
2
the above type. Hence
dim Bxui =
dim C u = (n − δ) +
n+δ−2 n−δ
−
= n − 1,
2
2
i.e. C u consists of components of Hu . But from Corollary 1.15 b) in Chapter I it
follows that for δ > 2 the hyperplane section Hu is irreducible. Taking into account
that each line in X passing through y is contained in TX,y , we can sum up the
preceding discussion in the following lemma.
4.5. Lemma. In the conditions of Lemma 4.4 suppose in addition that δ > 2.
Then Hu = X ∩ Lu = X ∩ T (Yu , X) = C u coincides with the closure of a union of
all lines lying in X and intersecting Yu .
4.6. We return to the case δ = 4, n ≡ 0 (mod 4). Let x be a generic point of
S
Hu . Put Dx = Yz , where z runs through the set of general points of the cone
z
S(x, Yu ). Since for a generic point y ∈ Yu and a generic point z ∈ hx, yi
Yz ∩ Lu ⊃ (TYz ,y ∩ Yz )
and
Yz ∩ Lu 3 x,
x∈
/ TYz ,y ∩ Yz ,
we see that Yz ⊂ Lu and therefore
Yu ⊂ Dx ⊂ Hu ,
(4.6.1)
where from the definition of Dx and the proof of Lemma 4.5 it follows that both
inclusions are strict. Since
dim Dx = (n − 4) + 4 − dim Yz ∩ Yu = dim Hu − (dim Yz ∩ Yu − 1),
138
VI. SCORZA VARIETIES
we conclude that dim Yz ∩ Yu = 2 and Dx is a divisor on Hu (arguing as in the
proof of Proposition 3.2 in Chapter IV, it is easy to show that Yz ∩ Yu = P2 ).
If x0 ∈ Dx is another generic point, then Dx0 = Dx . In fact, let a ∈ Dx be
a generic point, and let z 0 be a generic point of the line ha, x0 i. Then Yz0 is a
nonsingular four-dimensional quadric,
x0 ∈ Yz0 ,
Yz0 ∩ Yu 6= ∅,
a ∈ Yz0 ⊂ Dx .
(4.6.2)
Thus, varying x ∈ Hu , we obtain a one-dimensional family of divisors Dx ⊂ Hu .
From the proof of Lemma 4.5 it follows that the base of this family is an image
of the line hx, x0 i, where x, x0 is a general pair of points of Hu , so that the family
{Dx } is rational.
If x0 , . . . xi , 0 ≤ i ≤ k0 − 1 is a general collection of points of Dx and v is a
generic point of the i-dimensional linear subspace hx0 , . . . , xi i, then
Yv ⊂ Dx .
(4.6.3)
In fact, from (4.6.2) it follows that (4.6.3) holds for i = 1. On the other hand, the
proof of Theorem 1.8 of Chapter V shows that if a is a generic point ofSDx , w is a
generic point of S i−1 Dx , and v is a generic point of hx, wi, then Yv = Yz , where
z
z runs through the set of generic points of the cone S(a, Yw ). Thus (4.6.3) can be
easily proved by induction on i.
In particular, if u0 is a generic point of S k0 −1 Dx , then Yu0 ⊂ Dx ⊂ Hu0 , so that
Dx ⊂
\
Hu 0 .
(4.6.4)
u0 ∈S k0 −1 Dx
From (4.6.3) and formulas (1.5.4) and (1.5.8) in Chapter V it follows that
dim S k0 −1 Dx = dim S k0 −2 Dx + [(n − 1) − (n − 4)]
= dim S k0 −3 Dx + [(n − 1) − (n − 4)] + [(n − 1) − (n − 8)]
= · · · = (4k0 − 2) + 3 + 7 + · · · + [3 + 4(k0 − 2)] = 2k02 + k0 − 1.
By Theorem 1.4, the hyperplane Lu0 is tangent to S k0 −1 X along the linear subspace
Pu0 = S k0 −1 Yu0 ,
dim Pu0 = N − n − 1 = 2k02 − k0 − 1.
Hence, varying u0 ∈ S k0 −1 Dx , we obtain a [(2k02 + k0 − 1) − (2k02 − k0 − 1)] = 2k0 dimensional family of hyperplane sections Hu0 of the variety X passing through
the (n − 2)-dimensional subvariety Dx ⊂ Hu . It is clear that this family cuts a
(2k0 − 1)-dimensional family of hyperplane sections of the variety Hu containing
Dx as an irreducible component.
By Barth’s theorem (cf. [6; 33]), H 1 (X, OX ) = 0, and the exact sequence
0 → H 0 (X, OX ) → H 0 (X, OX (1)) → H 0 (Hu , OHu (1)) → H 1 (X, OX (1))
4. SCORZA VARIETIES WITH δ = 4
139
shows that the variety Hu ⊂ PN −1 is linearly normal (the Scorza variety X ⊂ PN
is linearly normal by definition).
Thus we obtain two linear systems I = |Dx | and II = |(Hu0 · Hu ) − Dx | on the
variety Hu whose fundamental subset coincides with Yu . We have already shown
that
dim I ≥ 1,
dim II ≥ 2k0 − 1.
(4.6.5)
Since Hu is linearly normal,
dim H 0 (Hu , OHu (1 − Yu )) = dim πu (Hu ) = n − 1 = 4k0 − 1
(4.6.6)
(here |[OHu (1 − Yu )]| is the linear system of hyperplane sections of Hu passing
through Yu ). From (4.6.5) and (4.6.6) it follows that
dim I = 1,
dim II = 2k0 − 1.
Furthermore, the linear system I maps Hu onto P1 , the linear system II maps Hu
onto P2k0 −1 , and the projection πu with center in Pu defined by the linear system
|[O(1 − Yu )]| maps Hu onto ¯P1 × P2k0 −1 . In the proof of Lemma 4.5 we actually
showed that the fibers of πu ¯Hu are cones
¡
¢
πu−1 πu (x) = Pu,x ∩ X = Cxu
with vertex x ∈ Hu \ Yu and base Bxu ,
dim Bxu =
dim Yu
= 2k0 − 2,
2
dim Cxu = 2k0 − 1.
Furthermore, arguing as in the proof of Proposition 3.2 in Chapter IV or using induction on n, it is easy to verify¯ that Bxu = Cxu ∩Yu is a linear subspace and therefore
the fibers of the projection πu ¯H are (2k0 − 1)-dimensional linear subspaces.
u
Summing up the above discussion, we get the following result.
4.7. Lemma. In the conditions of Lemma 4.5 suppose in addition that δ =
n
4. Then πu (Hu ) ⊂ πu (Lu ) ⊂ πu (PN ) = P1¯ × P 2 −1 ⊂ Pn−1 ⊂ Pn (the Segre
n
embedding), and a generic fiber of the map πu ¯H 99K P1 ×P 2 −1 is a linear subspace
u
n
of P 2 −1 .
4.8. Consider the map
σu = πu−1 : Pn 99K P
n(n+6)
8
,
σu (Pn ) = X.
From Lemma 4.7 it follows that σu is defined by a linear system of hypersurfaces
n
n
in Pn passing
through´ P1 × P 2 −1 ⊂ Pn−1 ⊂ Pn . Since P1 × P 2 −1 is defined in
³
¡
¢
n
2
Pn by (n + 1) + 22
= n +6n+8
= N + 1 quadratic equations, from the linear
8
normality of X it follows that σu is defined by the linear system of quadrics in Pn
n
passing through P1 × P 2 −1 .
Theorem 4.1 in the case n ≡ 0 (mod 4) now follows from the characterization of
Grassmannians (cf. [81] and also § 3 of Chapter III).
140
VI. SCORZA VARIETIES
4.9. Lemma. Let X n ⊂ PM (n,4) , δ(X) = 4 be a Scorza variety. Then n 6≡ 1
(mod 4).
Proof. Suppose the converse, and let u be a generic point of PN = S k0 X,
N = M (n, 4) = f (k0 ) =
n2 + 6n + 1
.
8
By Theorem 1.4, Yu = X ∩ Pu is a Scorza variety of dimension 4k0 = n − 1
2
in the linear subspace Pu = hYu i ⊂ PN , dim Pu = n +4n−5
< N − 1. Hence
8
for a general hyperplane L ⊃ Pu we have L · X > Yu which contradicts the BarthLarsen theorem according to which H2n−2 (X, Z) and Pic X are infinite cyclic groups
generated by the classes of hyperplane section (cf. [54; 65]). This contradiction
proves Lemma 4.9. ¤
4.10. We turn to the case n ≡ 2 (mod 4). Let
X n ⊂ PN ,
n ≡ 2 (mod 4),
N=
n(n + 6)
8
(4.10.1)
be a Scorza variety. According to Theorem 1.4, if u ∈ PN = S k0 X is a generic
point, then Yu = X ∩ Pu is a Scorza variety of dimension 4k0 = n − 2 in the linear
subspace Pu = hYu i ⊂ PN ,
dim Pu =
(n − 2)(n − 4)
=N−
8
n
2
− 1.
(4.10.2)
Yu ∩ Yu0 = Yu ∩ Pu0 ,
(4.10.3)
Let u0 ∈ PN be another generic point. Then
Yu0 = X ∩ Pu0 ,
and since dim Yu + dim Pu0 = (n − 2) + (N − n2 − 1) > N, we see that Yu ∩ Yu0 6= ∅.
From Theorem 1.4 it immediately follows that
δ(Yu ∩ Yu0 ) = δ(Yu ) = δ(X) = 4.
(4.10.4)
Furthermore, for a general pair of points u, u0 ∈ PN we have
dim Yu ∩ Yu0 = n − 4.
(4.10.5)
In fact, if we had
dim Yu ∩ Yu0 = dim Yu ∩ Pu0 = n − 3 = dim Yu − 1,
then Yu ∩ Yu0 would coincide with a hyperplane section of Yu which contradicts
(4.10.4).
From Theorem 1.4 and the already proven part of Theorem 4.1 it follows that
Yu = G(2k0 + 1, 1).
(4.10.6)
4. SCORZA VARIETIES WITH δ = 4
141
From (4.10.3), (4.10.4), (4.10.5), and (4.10.6) it is easy to deduce that
Y = Yu ∩ Yu0 = G(2k0 , 1)
(4.10.7)
is the Schubert cycle in Yu = G(2k0 + 1, 1) parametrizing the lines contained in the
hyperplane P2k0 ⊂ P2k0 +1 . It also easily follows that
Pu ∩ Pu0 = hY i = PY ,
(n−4)(n+2)
8
dim PY =
= N − n − 1.
Put
L = hPu , Pu0 i,
dim L = 2(N −
n
2
− 1) − (N − n − 1) = N − 1.
From the definition it follows that the hyperplane L is tangent to X along the
subvariety Y , i.e. T (Y, X) ⊂ L. Denote by H the hyperplane section X ∩ L and by
πY : X 99K Pn the projection with center at PY .
Varying a generic point u00 ∈ PN , we obtain an (N − dim Pu00 ) = ( n2 − 1)dimensional rational family of subvarieties Yu00 ⊂ X. Furthermore, if
Yu ∩ Yu00 = Yu ∩ Yu0 = Y,
(4.10.8)
then
hPu , Pu00 i = hPu , Pu00 i = L.
In this case
Yun−2
⊂ H n−1 ,
00
(4.10.9)
and it is clear that a generic point of H is contained only in a finite number of
subvarieties Yu00 satisfying condition (4.10.8). Since on the Grassmannian G(2k0 +
1, 1) there is a dim P(2k0 +1)∗ = (2k0 + 1) = n2 -dimensional family of Grassmannians
G(2k0 + 1), from (4.10.7) it follows that a general variety Y = Yu ∩ Yu0 = G(2k0 , 1)
is contained in a one-dimensional rational family of Yu00 and
[
H=
Yu00
(4.10.10)
Yu00 ∩Yu =Y
(cf. (4.10.9)). A word-for-word repetition of the arguments used in the proof of
Lemma 4.7 (with the linear system |Dx |, Yu ⊂ Dx ⊂ Hu replaced by the linear
system |Yu00 |, Y ⊂ Yu00 ⊂ H) shows that
¯
n
πY ¯H : H 99K P1 × P 2 −1 ⊂ Pn−1 ⊂ Pn = πY (X)
(4.10.11)
n
is a rational fiber bundle with fiber P 2 −1 .
We claim that for a generic point x ∈ X \ H
PY,x ∩ X = Y ∪ x,
(4.10.12)
where PY,x = hPY , xi. Suppose that this is not so, and let
y ∈ PY,x ∩ X,
y ∈ Y,
y 6= x.
(4.10.13)
142
VI. SCORZA VARIETIES
Fig. 3.
Put z = hx, yi ∩ PY , and let v be a generic point of PY and v 0 ∈ hz, vi a generic
point of the line hz, vi. Put
ũ = hx, vi ∩ hy, v 0 i
(cf. fig. 3).
It is easy to see that ũ is a generic point of PN . Since v, v 0 ∈ PY = S k0 −1 Y ⊂
S k0 −1 X,
x, y ∈ Yũ ⊃ Yv , Yv0 .
(4.10.14)
T
k0 −1
From (4.10.7) it follows that if Pw = S
Yw , then
Pw = ∅, and so we may
w∈PY
assume that
Yv0 6= Yv .
(4.10.15)
On the other hand,
Yũ ∩ Y = Yũ ∩ Yu ∩ Yu0 = (Yũ ∩ Yu ) ∩ (Yu0 ∩ Yu ).
(4.10.16)
Since x ∈
/ H, Yũ 6⊃ Y and as was shown above
¡ Yũ ∩ ¢Yu and Y = ¡Yu0 ∩
¢ Yu are
two distinct subgrassmannians of the form G n2 − 1, 1 in Yu = G n2 ,¡1 . From
¢
(4.10.16) it follows that Yũ ∩ Y is a Grassmann variety of the form G n2 − 2, 1 .
From (4.10.14) it follows that
Yũ ∩ Y ⊃ Yv , Yv0 ,
(4.10.17)
where in view of the already proven case of Theorem 4.1
¡ each of¢ the varieties
Yvn−6 , Yvn−6
is also a Grassmann variety of the form G n2 − 2, 1 (n − 6 ≡ 0
0
(mod 4)). Thus assumption (4.10.13) leads to a contradiction ((4.10.17) is incompatible with¯ (4.10.15)) which proves (4.10.12).
Thus πY ¯X\H is a birational isomorphism. Consider the inverse map
σY = πY−1 : Pn 99K P
n(n+6)
8
,
σY (Pn ) = X.
4. SCORZA VARIETIES WITH δ = 4
143
From the already proven case of Theorem 4.1 it follows that for a generic point
u ∈ PN
πY (Yu ) = Pu ⊂ Pn ,
dim Pu = n − 2,
and if Pn−1 = hπY (H)i, then
n
n
Pu ∩ Pn−1 = hP1 × P 2 −2 i,
n
P1 × P 2 −2 = πY (Yu ∩ H) ⊂ πY (H) = P1 × P 2 −1 .
Suppose that σY is defined by
n2 +6n+8
8
forms G0 , . . . , G n(n+6) , deg Gi = d, i =
8
0, . ¯. . , n(n+6)
. As it was shown in 4.8, after canceling the greatest common divisor
8
Gi ¯Pu become quadratic forms. Varying generic point u ∈ PN , we see that d = 2.
¡n¢
2
n
= N + 1 quadratic
Since P1 × P 2 −1 is defined in Pn by (n + 1) + 22 = n +6n+8
8
equations and
¯
n(n + 6)
i = 0, . . . ,
,
Gi ¯P1 ×P n2 −1 = 0,
8
from this it follows that σY is defined by the linear system of quadrics in Pn passing
n
through P1 × P 2 −1 .
Theorem 4.1 in the case n ≡ 2 (mod 4) now follows from the characterization of
Grassmann varieties (cf. [81] and § 3 of Chapter III).
It remains to consider the case n ≡ 3 (mod 4).
4.11. Lemma. Let X n ⊂ PM (n,4) , δ(X) = 4 be a Scorza variety. Then n 6≡ 3
(mod 4).
Proof. Suppose that the lemma does not hold, and let u be a generic point of
PN = S k0 X,
k0 =
n−3
,
4
N = M (n, 4) = f (k0 ) =
n2 + 6n − 3
,
8
and u0 a generic point of S(Yu , S k0 −1 X). Then u0 is a generic point of PN , and (by
Theorem 1.4) Yu and Yu0 are Scorza varieties of dimension n − 3. From the already
proven case of of Theorem 4.1 it follows that Yu and Yu0 are projectively isomorphic
n−1
to the Grassmann variety of lines in P 2 . Denote by Y the intersection Yu ∩ Yu0 .
It is clear that
dim Y ≥ n − 6,
δ(Y ) = 4,
(4.11.1)
and if z ¡∈ SY \¢ Y , then Yz ⊂ Y . Geometrically this means that if the subvariety
Y ⊂ G n−1
2 , 1 contains a pair of points α1 , α2 corresponding to non-coplanar
n−1
lines l1¡, l2 ⊂ P¢ 2 , then for each line l ⊂ hl1 , l2 i the subvariety Y contains a point
n−1
α ∈ G 2 , 1 corresponding to this line. From this it follows that
Y = G(m, 1),
n−1
(4.11.2)
m
2
where
is a linear subspace. It is clear that the only subvariety Y ⊂
¡ n−1P ¢ ⊂ P
G 2 ¡, 1 satisfying
conditions (4.11.1) and (4.11.2) is the Grassmann subvariety
¢
Y = G n−3
,
1
.
2
144
VI. SCORZA VARIETIES
We observe that if u0 ∈ hx, vi, where x ∈ Yu , v ∈ S k0 −1 X are generic points and
¡
¢
z is a generic point of Yv1 = p01,k0 −3 (ϕ1,k0 −3 )−1 (v) , then z is a generic point of
SX ,
½
µ
¶
µ
¶¾
n−5
n−1
{Yz ,→ Yv ,→ Yu0 } = G(3, 1) ,→ G
, 1 ,→ G
,1
,
2
2
½ µ
¶
µ
¶¾
n−3
n−1
{Y ,→ Yu0 } = G
, 1 ,→ G
,1
,
2
2
and therefore for a generic pair of points u ∈ S k0 −1 X, z ∈ SX
Yu ∩ Yz = G(2, 1) = P2 .
(4.11.3)
But from (4.11.3) it follows that
S(Yu , X) = SX,
dim ϕ−1
Y (z) = 2
while we already know that for a generic point z ∈ SX
dim ϕ−1
Yu (z) = dim Yu + n + 1 − dim SX = 1.
¤
Now all the four cases in Theorem 4.1 are verified, and the proof of the theorem
is complete. ¤
4.12. Remark. For δ = 4, n ≤ 7 we have k0 = 1, and each variety X n ⊂ P2n−3
is extremal. In other words,
M (7, 4) = m(7, 4) = 11,
M (6, 4) = m(6, 4) = 9,
M (5, 4) = m(5, 4) = 7,
M (4, 4) = m(4, 4) = 5.
4.13. Corollary.
Let X n ⊂ Pr be a nonsingular variety, r ≤ 2n − 3. Then
¡
¢ h (n+3)2 i
h X, OX (1) ≤
with equality holding if and only if r = 2n − 3 and either
8
0
r
n≤
¡ n7 or n ¢≡ 0 (mod 2) and X ,→ P is the embedding of the Grassmann variety
G 2 + 1, 1 ' X defined by a collection (Q0 : · · · : Q2n−3 ) of 2n − 2 linear forms
of the Plücker coordinates.
5. END OF CLASSIFICATION OF SCORZA VARIETIES
145
5. End of classification of Scorza varieties
It remains to classify Scorza varieties with δ = 8.
n
M (n,8)
, δ(X) = 8 be a Scorza variety. Then k0 =
£ n ¤5.1. Lemma. Let X ⊂ P
=
2.
8
Proof. Suppose that k0 ≥ 3, and let v be a generic point of S 3 X. From Theorem 1.4 it follows that Yv24 is a Scorza variety with δ(Y ) = 8. Hence to prove
Lemma 5.1 it suffices to verify that a 24-dimensional variety with δ = 8 does not
exist.
In fact, if X were such a variety, then we would have an ascending chain of secant
varieties
X 24 ⊂ (SX)41 ⊂ (S 2 X)50 ⊂ S 3 X = P51 .
Let u be a generic point of S 2 X. Put
Lu = TS 2 X,u ,
Hu = (Lu · X)P51 = Lu ∩ X.
Let x ∈ Hu be
¯ a generic point. In the proof of Lemma 4.5 it was shown that
Bxu = {y ∈ Yu ¯ hx, yi ⊂ X} is a dim2 Yu =8-dimensional subvariety of Yu (cf. (4.4.5),
(4.4.6)). As in 4.6, we see that Bxu = SBxu , and therefore Bxu is a linear subspace
(cf. the proof of Lemma 3.6 in Chapter IV). Hence in order to prove Lemma 5.1
it suffices to verify that the Severi variety Yu = E 16 ⊂ P24 does not contain eightdimensional linear subspaces.
We claim that E 16 actually does not contain even¡ six-dimensional
linear sub¢
spaces (we recall that for z ∈ SE \ E we have πz−1 πz (x) = P5 for each point
x ∈ Hz = Lz ∩ X, x ∈
/ Yz , so that E contains five-dimensional linear subspaces; cf.
also [13]). In fact, if E 16 ⊃ P6 3 x, then
TE∩x ∩ E ⊃ P6 .
(5.1.1)
In view of the results of §2 of Chapter III (cf. also Chapter IV, 4.2 c), 4.3), TE,x ∩
E is a cone with vertex x and base S 10 , where S 10 ⊂ P15 is the spinor variety
corresponding to the orbit of highest weight vector of the spinor representation of
the group Spin10 . Hence from (5.1.1) it follows that S 10 ⊃ P5 . But for an arbitrary
point y ∈ P5 ⊂ S 10
TS,y ∩ S ⊃ P5
(5.1.2)
is a cone with vertex y and base G(4, 1) ⊂ P9 (cf. §2 in Chapter III). Hence from
(5.1.2) it follows that G(4, 1) ⊃ P4 . But for an arbitrary point α ∈ P4 ⊂ G(4, 1)
TG(4,1),α ∩ G(4, 1) ⊃ P4
(5.1.3)
is a cone with vertex α and base P1 × P2 ⊂ P5 (cf. again §2 of Chapter III). Hence
(5.1.3) would imply that P1 × P2 ⊃ P3 which is clearly impossible.
This completes the proof of Lemma 5.1. ¤
From Lemma 5.1 it follows that the dimension n of a Scorza variety X n with
δ(X) = 8 satisfies the inequalities
16 ≤ n ≤ 23.
146
VI. SCORZA VARIETIES
5.2. Lemma. Let X n be a Scorza variety such that δ(X) = 8 and k0 (X) = 2.
Then n = 16 and X = E is a Severi variety.
Proof. Suppose that 17 ≤ n ≤ 23. Then N = M (n, 8) = f (2) = 26 + 3ε,
ε = n mod 8, and for a generic point u ∈ S 2 X = PN
© 16
ª ©
ª
Yu ⊂ P26 = E 16 ⊂ P26
is a sixteen-dimensional Severi variety (cf. Theorem 1.4). Let u0 be a generic point
of S(Yu , SX). Then u0 is a generic point of PN , and Yu0 is a sixteen-dimensional
Severi variety.
Put Y = Yu ∩ Yu0 . It is clear that Y is a nonsingular variety,
9 = 32 − 23 ≤ dim Y ≤ 15,
(5.2.1)
and if z ∈ SY \ Y , then Yz ⊂ Y (cf. the proof of Lemma 4.11). From (5.2.1) and
Corollary 2.11 in Chapter II it follows that
SY = Pr ,
and therefore
r = 2 dim Y − 7 ≥ 11,
SE ⊃ P11 .
(5.2.2)
But according to the results of §2 of Chapter III and Remark 2.5 in Chapter IV
(SE)∗ ' E and each hyperplane TSE,z (z ∈ SE \ E) is tangent to SE along the
linear subspace P9z = hYz i. Since for z ∈ P11 \ E
TSX,z ⊃ P11 ,
(5.2.3)
from (5.2.2) and (5.2.3) it follows that P11 ⊂ E. But in the proof of Lemma 5.1 we
verified that the variety E does not contain even six-dimensional linear subspaces.
The resulting contradiction proves Lemma 5.2 (the non-existence of Scorza varieties
X n with 17 ≤ n ≤ 19 can be more easily deduced from the Barth-Larsen theorem
[54], but we prefered to give a more uniform proof). ¤
Combining the assertions of Lemmas 5.1 and 5.2 and Theorem 4.7 in Chapter IV
we obtain the following result.
5.3. Theorem. Let X n ⊂ PN , n ≥ 16 be a nonsingular nondegenerate variety.
Suppose that sX < 2n − 6. Then N ≤ n(n+10)+ε(6−ε)
, ε = n mod 4. Furthermore,
16
equality holds if and only if X = E ⊂ P26 is a sixteen-dimensional Severi variety
corresponding to the orbit of highest weight vector of the simplest representation
of the group E6 . In other words, M (16, 8) = 26 and M (n, 8) < n(n+10)+ε(6−ε)
for
16
n > 16.
5.4. Remark. For δ = 8, 8 ≤ n ≤ 15 we have k0 = 1 and each variety X n ⊂
P
is extremal. In other words,
2n−7
M (n, 8) = m(n, 8) = 2n − 7,
8 ≤ n ≤ 15.
5. END OF CLASSIFICATION OF SCORZA VARIETIES
147
n
r
5.5. Corollary.
h Let2 iX ⊂ P be a nonsingular variety, r ≤ 2n − 7. Then
(n+5)
h0 (X, OX (1)) ≤
with equality holding if and only if r = 2n − 7 and
16
either n ≤ 14 or n = 16 and X ⊂ P25 is an isomorphic projection of the sixteendimensional Severi variety E 16 ⊂ P26 .
Combining Theorems 2.1, 3.1, 4.1, and 5.3 and taking into account properties of
the function f depicted on fig. 3 in Chapter V, we obtain the following
5.6. Classification theorem. Let X n ⊂ PN be a nonsingular nondegenerate variety over an algebraically closed field K. Then N ≤ n(n+δ+2)+ε(δ−ε−2)
≤
h
i
© n ª 2δ
(n+ δ2 +1)2
− 1, where δ = 2n + 1 − s, s = dim SX, ε = δ δ = n mod δ. If
2δ
char K = 0, then N = n(n+δ+2)+ε(δ−ε−2)
in the following cases:
2δ
(0) n < 2δ, ε = n − δ, s = N = 2n + 1 − δ;
, X = v2 (Pn ) is the Veronese variety;
(i) δ = 1, N = n(n+3)
2
n+1
n
(ii) δ = 2, N = n(n+4)−nmod2
, X = P[ 2 ] × P[ 2 ] is the Segre variety;
4
¡
¢
, X = G n2 + 1, 1 is the Grassmann
(iii) δ = 4, n ≡ 0 (mod 2), N = n(n+6)
8
variety;
(iv) δ = 8, n = 16, N = 26, X = E is the sixteen-dimensional Severi variety.
h
i
(n+ δ2 +1)2
The varieties (i)–(iv) are Scorza varieties, and for them N =
− 1.
2δ
n
The Scorza variety X corresponds to the orbit of highest weight vector of an
irreducible representation of a semisimple group G in a vector space V with highest
weight Λ, where
(i) G = SLn+1 ,
Λ = 2ϕ1 ;
(ii) G = SL[ n+2 ] × SL[ n+3 ] ,
Λ = ϕ1 ⊕ ϕ1 ;
2
2
(iii) G = SL n2 +2 ,
Λ = ϕ2 ;
(iv) G = E6 ,
Λ = ϕ1
(here ϕi is the i-th fundamental weight).
The Scorza variety X n is the image of Pn under the rational map σ : Pn 99K PN
defined by the linear system of quadrics passing through the subvariety A ⊂ Pn−1 ⊂
Pn , where
(i) A = ∅;
n−2 `
n−1
(ii) A = P[ 2 ] P[ 2 ] ;
n
(iii) A = P1 × P 2 −1 ;
(iv) A = S 10 .
In other words, the Scorza variety X n is obtained from the Veronese variety
n(n+3)
v2 (Pn ) ⊂ P 2
by projecting from the linear span hv2 (A)i of the image of the
variety A ⊂ Pn under the Veronese map v2 .
5.7. Remark. As in Remark 4.3 in Chapter IV, it is not hard to verify that for
an arbitrary point x of the Scorza variety X n ⊂ P¯N the variety TX,x ∩ X is a cone
with vertex x and base A. Under the map π = (σ ¯X )−1 each of the cones TX,x ∩ X
is mapped onto its base. The map π is an isomorphism outside of σ(Pn−1 ) and
[
TX,y ∩ X,
π : σ(Pn−1 ) → A.
σ(Pn−1 ) =
y∈Sing σ(Pn−1 )
148
VI. SCORZA VARIETIES
5.8. Remark. As in Remark 4.6 in Chapter IV, it is not hard to verify that
the linear system of quadrics cut in a general linear subspace Pn−1 ⊂ TX,x by
the linear system of quadrics passing through the Scorza variety X and defining a
x
M (n−δ,δ)
rational¡map Pn−1
(where Y x ⊂ X ∗ is naturally isomorphic
¢ 99K Y ⊂ P
n−1
to Sing σ(P
) ) is the second fundamental form in the sense of [29] and the
subvariety A ⊂ Pn−1 is the fundamental subset of this form.
Combining Corollaries 2.9, 3.5, 4.13, and 5.5 with Theorem 2.10 of Chapter V
we obtain the following.
5.9. Theorem. Let X n ⊂ Pr be ha nonsingular
i variety over an algebraically
(4n−r+3)2
0
closed field K. Then h (X, OX (1)) ≤ 8(2n−r+1) . If in addition char K = 0 and
r ≥ 3n
2 + 1, then equality holds if and only if X is an isomorphic projection of
a Scorza variety X̃ to a projective space Ps , s = dim S X̃, so that in particular
3n
r = 2n, 2n − 1, 2n − 3 or 2n
¡ − 7 (if ¢r < 2 + 1, then from Corollary 2.11 in
0
Chapter II it follows that h X, OX (1) = r + 1).
5.10. Remark. It is worthwhile to observe that in the most important case
when n ≡ 0 (mod δ) classification of Scorza varieties over an algebraically closed
field K, char K = 0 is parallel to classification of Jordan
¡ matrix
¢ algebras due to
Albert (cf. [10; 44, Chapter V ]). More precisely, PN = P J nδ +1 where J nδ +1 is the
Jordan algebra of Hermitean matrices of order nδ + 1 over a composition algebra A,
dimK J nδ +1 = (n+δ)(n+2)
(we recall that for A = A0 , A1 , A2 nδ ≥ 2 is an arbitrary
2
integer and J nδ +1 is a special Jordan algebra and for A = A3 we have nδ = 2
and J3 is an exceptional©Jordan algebra;
cf. Theorem
4.8 in Chapter III) and X
¯
ª
corresponds to the cone A ∈ J nδ +1 ¯ rk A ≤ 1 . More
generally,
the variety S k X,
¯
©
ª
n
¯
n
0 ≤ k ≤ k0 = δ corresponds to the cone A ∈ J δ +1 rk A ≤ k + 1 .
Arguing as in Theorem 4.9 of Chapter IV, we can restate Remark 5.10 as follows.
n
N
5.11. Theorem.
¡ n ¢ A nonsingular nondegenerate variety X ⊂ P , n ≡ 0
(mod δ), N = f δ , δ = δ(X) over an algebraically closed field K, char K = 0
is a Scorza variety if and only if X is the ‘Veronese variety’ of dimension nδ ≥ 2
over the composition
¡ n algebra
¢ A, dimK A = δ, i.e. X is the image of the ‘projective
n
space’ P δ (A) = A δ +1 \ 0 /A∗ (where A∗ is the subset of invertible elements of
v2
the algebra A) under the map (x0 : · · · : x nδ ) 99K (· · · : xl x̄m : . . . ), 0 ≤ l ≤ m ≤ nδ
(for δ = 8¡ we have¢ n = 16 since, due to the lack of associativity, for larger n the
n
variety v2 P¡ 8 (A3 )¢ is ¡no longer
¢ defined by vanishing of the minors of order two of a
Hermitean n8 + 1 × n8 + 1 -matrix; this corresponds to J3 being the only Jordan
matrix algebra over A3 ).
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