Download LINE BUNDLES AND DIVISORS IN ALGEBRAIC GEOMETRY

LINE BUNDLES AND DIVISORS IN ALGEBRAIC GEOMETRY
JOHN KOPPER
Contents
1. Introduction
2. Line bundles and invertible sheaves
3. Divisors
References
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1. Introduction
It has been said that Chapter II, §§6-7 in Hartshorne [3] (Divisors and Projective Morphisms, respectively) are among the most important parts of the book.
Unfortunately, the treatment of the subject is inscrutably abstract, especially for
someone struggling to understand schemes (like me). In writing these notes my
hope is to provide intuitive, if not rigorous, discussions of these topics to convince
myself and perhaps an occasional reader that they are, in fact, important.
Essentially all of the contents of these notes are taken directly from the references. Hartshorne’s Algebraic Geometry [3] is the definitive text on the subject,
and the original impetus for these notes. Section 2 in particular is essentially a
long-winded version of Exercise II.5.18. Vakil’s Foundations of Algebraic Geometry
[4] provides an alternate perspective while still using the formalism of schemes. In
particular, Vakil defines line bundles through transition functions, an idea which I
have found useful in my own understanding. Griffiths and Harris’ [1] book Principles of Algebraic Geometry is written from the less general perspective of complex
varities, which are a very nice special case of many ideas. I recommend this last for
motivation and intuitive understanding, while I recommend the first two for their
exercises and for extending beyond the complex case.
Because our goal is to understand, not to prove, many of the proofs in these
notes are incomplete or not rigorous. The reader who wants to see these things will
find them in any of the above mentioned references.
2. Line bundles and invertible sheaves
I will begin by stressing that line bundles on a variety and invertible sheaves on
that variety are not literally the same thing. On the other hand, they correspond
to each other in a natural manner. It is thus common for people to use the terms
interchangeably. Let’s start with line bundles.
Definition 2.1. Let X be a variety. Then a vector bundle of rank n on X is a
space E and a map π : E → X such that
(1) π −1 (x) is an n-dimensional k-vector space.
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2
JOHN KOPPER
(2) For each x ∈ X there exists an open set U containing x so that U ×k n ∼
= π −1 (U )
via a homeomorphism ϕU , called a local trivialization.
(3) π(ϕU (x, v)) = x and v 7→ ϕU (x, v) is a vector space isomorphism k n → π −1 (x).
A vector bundle of rank one is called a line bundle.
Definition 2.2. Let π : E → X be a vector bundle, and denote π −1 (x) = Vx . A
section of π is a map s : X → E so that (π ◦ s)(x) = x, i.e. so that s(x) ∈ Vx . If
U ⊆ X is a subset we can refer to sections of U . Sections of X are called global
sections.
Example 2.3 (Trivial bundle). The easiest line bundle to associate to a space X
is the trivial bundle X × k with the map π : X × k → X simply projection to the
first factor. Let’s explore this in the case X = Pn . Then E = X × k; what are the
sections of this line bundle?
If s is a section of this line bundle then we get a map s0 : Pn → k given by s0 (x) =
π2 (s(x)), where π2 : X × k → k is projection to the second factor. Proposition 2.21
will show that s must be a constant map. Thus the sections of this line bundle are
the constant maps.
If U ⊆ X and s, t : U → E are sections of a vector bundle π : E → X then
defining (s + t)(x) = s(x) + t(x), (s · t)(x) = s(x)t(x), (α · s)(x) = αs(x) for α ∈ k
gives a k-algebra structure to the sections of U . Example 2.3 shows that global
sections of Pn are isomorphic to k.
To produce more sophisticated examples of vector bundles we’ll need some tools
for describing them. For example, the Möbius band is a nontrivial line bundle
over the circle. This is easy to picture, but hard to write symbolically. This next
definition will give us a way of describing how to “glue” the local trivializations
together.
Definition 2.4. Let π : E → X be a vector bundle, and U and V open sets
containing x so that U × k n ∼
= π −1 (V ) via local trivializations
= π −1 (U and V × k n ∼
ϕU and ϕV respectively. Then there is a map
ϕU V : U ∩ V → GL(k n )
given by ϕU V (x) = (ϕ−1
V ◦ ϕU )|{x}×kn .
This is a fancy way of saying that two local trivializations in a neighborhood of
a point have to fit together via an isomorphism in that neighborhood. It is a fact,
and not a difficult one, that the transition functions determine the vector bunde
and vice-versa (see, for example, [1] p. 66).
Remark 2.5. Notation for transition functions is shockingly inconsistent in the
literature. Many authors also have their local trivializations go the other way, i.e.
ϕU : π −1 (U ) → U × k n . I have done my best to keep the notation consistent
through this document.
Example 2.6 (Tautological bundle). Consider projective space Pn again. By definition, Pn is the set of all lines through the origin in k n+1 . We can thus construct
a line bundle on Pn by associating to every point x the corresponding line in k n+1 .
What does the total space look like?
To answer this question, we’ll use transition functions. On an affine neighborhood Ui = {[Z0 : · · · : Zn ] : Zi 6= 0} we have π −1 (Ui ) ∼
= Ui × k, and the point
LINE BUNDLES AND DIVISORS IN ALGEBRAIC GEOMETRY
3
z = [Z0 : · · · : Zn ] corresponds to the line spanned by the vector (Z0 , . . . , Zn ).
Assume Z0 6= 0 and Z1 6= 0 so that z ∈ U0 ∩ U1 .
Working in U0 , we can divide all coordinates by Z0 so that we have z = [1 :
Z1 /Z0 : · · · : Zn /Z0 ] and in U1 we have z = [Z0 /Z1 : 1 : · · · : Zn /Z1 ]. Thus the
transition function ϕU0 U1 must send the line spanned by (1, Z1 /Z0 , . . . , Zn /Z1 ) to
the line spanned by (Z0 /Z1 , 1, . . . , Zn /Z1 ), which is to say that ϕU0 U1 is multiplication by Z1 /Z0 .
We thus see that the tautological bundle on Pn doesn’t look like the trivial bundle. The affine neighborhoods glue together in a funny twisted way, somewhat
analogous to the Möbius band. Let’s compute the global sections. A global section must be given by a an element f ∈ k[Z0 /Z1 , Z2 /Z1 , . . . , Zn /Z1 ] on U1 and
g ∈ k[Z1 /Z0 , . . . , Zn /Z0 ] on U0 such that f = Z1 /Z0 g (and similar for the other
trivializations).
But this is impossible: if g is a nonzero polynomial then multiplication by Z1 /Z0
merely introduces more Z0 terms into the denominator. Thus there can be no
nonzero global sections of the tautological bundle on Pn .
A more standard notation that we will adopt is that ϕij denotes the transition
function from the affine open set Ui to Uj as in the above example.
Example 2.7. The transition functions in Example 2.6 give us a hint as to how
to construct another line bundle. Using our new notation, let E be the line bundle
over Pn with transition functions given by ϕij = Zi /Zj , i.e. the reciprocal of the
transition functions of the tautological bundle.
Let’s compute the global sections. A global section is given by polynomials f0 ∈
k[Z1 /Z0 , . . . , Zn /Z0 ], . . . , fn ∈ k[Z0 /Zn , . . . , Zn−1 /Zn ] such that fi Zi /Zj = fj . By
counting degrees it’s clear that the fi must be linear. Indeed fix i and set
X Zj
fi = b +
aj .
Zi
j6=i
Then the fj are determined, so we see that the dimension of global sections (as a
k-vector space) is n + 1.
Examples 2.3, 2.6 and 2.7 form our working toolbox of examples. We will see
(spoiler alert) that they correspond to the invertible sheaves O, O(−1) and O(1)
respectively.
Definition 2.8. Let X be a variety and OX its structure sheaf. A sheaf F on X
is called an OX -module if for each open set U ⊆ X, F (U ) is an OX (U )-module
compatible with the restriction morphisms. We say F is locally free of rank n if
there exist covering sets Uα such that F (Uα ) ∼
= O(Uα )⊕n for all α.
There is a natural product on locally free sheaves given by the tensor product.
Explicitly, if F and G are locally free sheaves on X then we define F ⊗ G to be
the sheaf associated to the presheaf
U 7→ F (U ) ⊗OX (U ) G (U ).
Proposition 2.9. If F and G are locally free sheaves of rank one on X then
F ⊗ G is too. Further, there exists a locally free sheaf F −1 of rank one such that
F ⊗ F −1 = OX .
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JOHN KOPPER
Proof. By definition F ⊗ G is the sheaf associated to the presheaf
U 7→ F (U ) ⊗OX (U ) G (U )
But on any such U we have F (U ) ∼
= G (U ) ∼
= OX (U ) and OX (U ) ⊗OX (U ) OX (U ) ∼
=
OX (U ). Thus on U F ⊗ G is free of rank one.
Let Fˇ = H omOX (F , OX ), where H omOX (F , OX ) is the sheaf associated with
the presheaf
U 7→ HomOX (U ) (F (U ), OX (U )).
Now for any OX -modules F and G we have an isomorphism
F ⊗ H omOX (F , G ) → G
given by m⊗f 7→ f (m). Indeed on stalks this map is just a map of Ox (U )-modules,
for suitable U , and is certainly an isomorphism of modules. In the case G = OX
we obtain the desired result.
The proposition implies that locally free sheaves of rank one form a group which
we call the Picard group of X, henceforth denoted Pic X. Since only the rank one
locally free sheaves are invertible, we get the following definition:
Definition 2.10. A locally free sheaf of rank one is called an invertible sheaf.
The obvious example of an invertible sheaf is OX itself. Other examples are not
so easy to produce.
Example 2.11 (Twisted sheaves). Let X ⊆ Pn be a projective variety and S =
S(X) its homogeneous coordinate ring. Then S is a graded ring. For an open set
U ⊆ X, let D(U ) = {f : f (x) 6= 0 for all x ∈ U }. We define a sheaf OX (m) (or
O(m)) by letting O(m)(U ) be the degree m elements of the localization D(U )−1 S.
Intuitively, if local sections of O look like rational functions f /g with deg f =
deg g then local sections of O(m) look like rational functions f /g with deg f =
deg g + m. I claim O(m) is an invertible sheaf.
Notice first that O(U ) is just O(0)(U ), so O(m) is certainly an O-module. Let
U = D(f ) be an open set which is the complement of the zero locus of some
polynomial f . Then we define a map O(m)(U ) → O(U ) by g 7→ g/f m . This is
clearly an isomorphism, thus O(m) is locally free of rank one.
Proposition 2.12. With notation as in Example 2.11, O(l) ⊗ O(m) = O(l + m).
Proof. This is relatively simple: on an open set U = D(f ) we have that elements
of O(l) are of the form g/f d where deg(g) − d = l and elements of O(m) are of the
form h/f e where deg(h) − e = m. Then their product is of the form gh/f d+e and
deg(g) + deg(h) − d − e = l + m.
We thus have a list of many invertible sheaves, though we haven’t shown they
are all different (and they need not be, depending on X). This next example shows
that an invertible sheaf corresponds in a natural manner to a line bundle.
Example 2.13. Let X ⊆ Pn be a variety with structure sheaf O = OX , and let E
be an invertible sheaf on X. Then let U and V be open subsets of X so that E (U )
and E (V ) are free. Then the restrictions E (U )|U ∩V and E (V )|U ∩V are isomorphic
as O(U ∩ V )-modules.
That is, we have O(U ∩ V )-module isomorphisms
ϕU : E (U )|U ∩V −→ O(U ∩ V )
ϕV : E (V )|U ∩V −→ O(U ∩ V )
LINE BUNDLES AND DIVISORS IN ALGEBRAIC GEOMETRY
5
hence an isomorphism ϕV ◦ ϕ−1
U : O(U ∩ V ) → O(U ∩ V ). This is just multiplication
by some element F ∈ O(U ∩ V ); evaluating F at a point z ∈ O ∩ V gives us an
element of k. Thus we define a line bundle π : E → X with transition maps ϕU V (z)
given by multiplication by F (z).
Let’s see 2.13 in the case X = Pn and E = O(1). We cover X by the affine Ui as
usual. We recall that sections of O(1) on Ui are of the form f /Zim where deg(f ) =
m + 1. The isomorphisms ϕUi and ϕVj are division by Zi and Zj , respectively, so
the isomorphism F is multiplication by Zi /Zj . Observe that these are precisely the
transition maps we found in Example 2.7. Since a vector bundle is determined by
its transition functions, we know it must be the same one!
Remark 2.14. The construction in example 2.13 works for E any locally free sheaf
of rank r. The isomorphism F is an isomorphism of free modules of rank r, hence
an r × r matrix. Evaluation of each entry at the point z gives an element of GLr (k).
Definition 2.15. Let E be a locally free sheaf on X. Denote by V(E ) the vector bundle attained using the process of Example 2.13, called the vector bundle
associated to the sheaf E .
Example 2.16. We have seen that V(O(1)) is the sheaf described in Example
2.7. It is also easy to verify that V(O) is the trivial bundle of Example 2.3, and
V(O(−1)) is the tautological bundle of Example 2.6.
We now would like to perform an inverse operation: take a line bundle and use
it to produce an invertible sheaf.
Definition 2.17. Let X ⊆ Pn be a variety and π : E → X a vector bundle. We
define the sheaf of sections LE to be the sheaf associated to the presheaf
U 7→ {s : X → E : π ◦ s = id}.
That is, the presheaf whose sections on U are the sections of π on U .
Remark 2.18. It is worth mentioning that these things can be confusing when
we leave out some of the rigor. In the definition we of course don’t mean all maps
X → E, we mean morphisms of schemes. But we are avoiding these words.
Proposition 2.19. If π : E → X is a vector bundle of rank r then LE is a locally
free sheaf of rank r.
Proof. Let Ui be open subsets of X so that π −1 (Ui ) ∼
= Ui × Ar via a local trivialization ϕUi . If s is a section of Ui we obtain a map Ui → An as the composition
Ui
/ π −1 (U )
ϕUi
/ Ui × Ar
π2
/ Ar .
Since Ar ∼
= A1 × · · · × A1 , a section of Ui is an r-tuple of maps to A1 , thus a free
O(Ui )-module of rank r.
We thus have a correspondence
V(·)
{locally free sheaves of rank r} o
/
{vector bundles of rank r}
L·
It would be nice if the maps V(·) and L· were mutual inverses. This is indeed the
case.
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JOHN KOPPER
Proposition 2.20. Let X ⊆ Pn be a variety. Let F be a locally invertible sheaf
on X and π : E → X a vector bundle. Then:
(1) LV(F ) = F .
(2) V(LE ) = E.
Proof. To show that two sheaves are isomorphic it suffices to show that they are
isomorphic locally and “agree appropriately” on intersections of open sets. We
have already seen that LV(F ) and F are locally isomorphic to O(U )r so they
are certainly locally isomorphic. Further, if U , V and W are open sets on which
they are free, then we obtain isomorphisms ϕU V , ϕU W , ϕV W which satisfy ϕU W =
ϕV W ◦ ϕU V . This is a property of transition functions and directly translates to
the corresponding property of sheaves. Thus (1).
If ϕU V is a transition function for local trivializations ϕU and ϕV of E then a
section s of U restricts to a section of U ∩ V and becomes a section of V restricted
r
r
to U ∩ V as ϕU V (s). Thus the isomorphism OU
∩V → OU ∩V given by LE (U )|U ∩V
and LE (V )|U ∩V is just the transition function ϕU V . Thus the transition functions
of V(LE ) and E are the same, hence (2).
We have omitted quite a few details in this proof. For the “glueing” property of
sheaves and bundles, see [3] p. 69. For the corresponding property of bundles, see
[1] p. 66 or [4] 13.1.1. We will henceforth drop the cumbersome notation V(·) and
L· and simply refer to the corresponding vector bundle or locally free sheaf.
Let us now consider maps from Pn and maps to Pn . We recall that to define a
projective subvariety of Pn we can specify a list of homogeneous polynomials and
take their common zero locus. Of course, the reason these polynomials must be
homogeneous is because the “zero locus” of a non-homogeneous polynomial doesn’t
make sense on projective space. Clearly for a polynomial to vanish at [Z0 : · · · : Zn ]
and [λZ0 : · · · : λZn ] it must be homogeneous.
In a similar vein, a map f : Pn → k must be a rational function whose numerator
and denominator are of the same degree. Indeed, if the numerator has degree d
and the denominator degree e, then evaluating f at z = [λZ0 : · · · : λZn ] gives
λd−e f (z) = f (z). Further, if f is to be defined on all of Pn then the denominator
cannot vanish. But if k is algebraically closed then the denominator necessarily
vanishes somewhere so long as its degree is nonzero. We conclude that a map
Pn → k is simply a constant function (and certainly any constant function is a
map). More formally,
Proposition 2.21. O(Pn ) = k.
Of course it’s true in greater generality that O(X) = k for X ⊆ Pn a projective
variety (see [3] I.3.4). But for our purposes, we care only about Pn , and we happily
conclude that we understand all maps Pn → k.
Maps to Pn are a bit more complicated. They are maps which are locally given
as homogeneous polynomials of the same degree. Pn is a bit special in this respect
because we can drop the word locally. Of course, the homogeneous polynomials
cannot all vanish simultaneously because [0 : · · · : 0] is not a point of Pn .
More explicitly, let X ⊆ Pm and let F : X → Pn be a morphism given by F =
[f0 : · · · : fn ]. The maps fi are then maps X → A1 , not vanishing simultaneously.
Let’s define a line bundle on X related to the fi : if fi is nowhere zero on an open
set Ui then we let Ui × A1 be a trivialization. If Ui and Uj are two such sets, define
a transition function ϕUi Uj by multiplication by fj /fi .
LINE BUNDLES AND DIVISORS IN ALGEBRAIC GEOMETRY
7
Taking the corresponding invertible sheaf L , we see that the global sections {fi }
generate, that is, if x ∈ X then the stalk Lx is generated by any of the fi which do
not vanish at x (because they are invertible there). Thus given a map F : X → Pn
we obtain a line bundle L and a collection of generating global sections {fi }.
Conversely suppose we have a line bundle L and generating sections f0 , . . . , fn .
Then we can define a map F : X → Pn with F = [f0 : · · · : fn ]. Since the fi
generate, they do not simultaneously vanish anywhere on X. Further, the transition functions ϕUi Uj are multiplication by an element gij of O(Ui ∩ Uj ). That is,
fj |Ui ∩Uj = gij fi |Ui ∩Uj . Since gij has degree 0, it follows that deg fi = deg fj , so
the map F is well-defined. In summary,
Proposition 2.22. Let X be a variety and F : X → Pn a morphism. Then
F ∗ ((O(1)) is an invertible sheaf on X which is generated by global sections F ∗ (Zi ),
and if F is an invertible sheaf on X generated by global sections s0 , . . . , sn then
there is a unique morphism F : X → Pn so that F = F ∗ (O(1)) and si = F ∗ (Zi ).
3. Divisors
There are two kinds of divisors we will consider: Weil divisors and Cartier divisors. The former are geometric in nature, and the latter are sheaf-theoretic. We
will see that the Cartier class group CaCl X is in general isomorphic to the Picard
group Pic X, and under somewhat special circumstances it is isomorphic to the
divisor class group Cl X.
The study of Weil divisors is the study codimension one subvarieties of a given
variety. This is a reasonable thing to do for a number of reasons. The most geometric one is that if X ⊆ Pn then X is an intersection of hypersurfaces (indeed if
the ideal of X is given by (f1 , . . . , fk ) then X is the intersection of the hypersurfaces cut out by the fi ). Thus understanding X is the same as understanding the
hypersurfaces containing it. Divisors also give us a convenient algebraic invariant
(a group) to work with.
Definition 3.1. If X is a variety then a prime divisor on X is a codimension one
(irreducible) subvariety. A Weil Divisor on X is an element of the free abelian
group Div X generated by the prime divisors of X.
P
Following standard notation, we will write a divisor D as a sum i ni Yi where
the ni are integers with all but finitely many equal to zero. If each of the ni ≥ 0 then
D is called effective. Given a function f on X, i.e. an element of the homogeneous
coordinate ring of X, we can associate a Weil divisor (f ) (this notation is standard
and unfortunate). To do so, we’ll start in the Hartshorne-esque way, then connect
it to our geometric intuition.
Definition 3.2. If X is a variety and Y ⊆ X a subvariety we define the local ring
of the subvariety Y OY,X to be the set of equivalence classes hU, f i such that U is
an open subset of X with nonempty intersection with Y , and hU, f i is equivalent
to hV, gi if f = g on U ∩ V .
Certainly OY,X is a local ring. If its maximal ideal is m and dim m/m2 = 1 for
all codimension one subvarieties then we say that X is regular in codimension one.
Intuitively this means that singularities on X can occur only in dimensions lower
than dim X − 1.
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JOHN KOPPER
Proposition 3.3. If X is regular in codimension one and Y ⊆ X is a codimension
one subvariety then OY,X is a discrete valuation ring with valuation vY .
Now if f is a rational function defined on X then its image in OY,X has a
valuation vY (f ). We thus define the divisor associated to f as
X
(f ) =
vY (f ) Y.
Y
Before we check that this is well-defined (i.e. that vY is zero for all but finitely
many Y ), let’s try to understand the valuation. It is a fact from commutative
algebra that if X is regular of codimension one then Y is given locally as the zero
locus of a single function π, and the corresponding local ring has maximal ideal
(π). Write the rational function f as g/h. We may therefore write g = uπ r and
h = vπ s where u and v are units. Then vY (f ) = r − s, which is to say the “order
of the zero” of f along Y .
Example 3.4. We haven’t made these ideas rigorous but it is nice to compute an
example to see what is happening. Let X = A2 with coordinates x, y and let Y be
the zero locus of the equation x = 0. Let f be the (global) rational function
x2 y
.
x
Then vY (f ) = 1. It is not always this easy: if Y is not a complete intersection then
we actually have to work locally to determine the valuation.
f (x, y) =
Example 3.5. Let X = Pn . We are in the fortunate situation that Y ⊆ Pn is a
hypersurface if and only if it is the zero locus of a single function f ∈ k[Z0 , . . . , Zn ].
Write f = f1n1 · · · frnr with each fi irreducible. Then the zero locus of fi is an
irreducible hypersurface Yi of Pn . Then the image of fi generates the maximal
ideal of OYi ,X . Since the Yi are the only prime divisors on which f vanishes, we
conclude
X
(f ) =
ni Yi .
i
Proposition 3.6. Let X be a variety regular in codimension one, and let f be a
rational function on X. Then vY (f ) = 0 for all except finitely many prime divisors
Y on X.
Proof. See Hartshorne [3] p. 131 [the proof is actually totally reasonable and you
don’t have to think about schemes at all].
Now that everything makes sense (or close enough), we can define the invariant
we are interested in.
Definition 3.7. Let X be a variety. Then a divisor D on X is called a principal
divisor if D = (f ) for some rational function on X. The group Div X modulo the
subgroup of principal divisors is called the divisor class group of X and is denoted
Cl X.
P
Example 3.8. Let’s consider the most obvious case: X = Pn . Let D = i ni Yi
be a divisor. Then we can write D = D1 − D2 where D1 and D2 are effective
divisors. Using the method of Example 3.5, we can produce functions f1 and f2 so
that D1 = (f1 ) and D2 = (f2 ).
LINE BUNDLES AND DIVISORS IN ALGEBRAIC GEOMETRY
9
P
Now let H be the hyperplane Z0 = 0, and let d = i ni deg Yi . Consider the
function g = f1 /(Z0d f2 ): the corresponding divisor is given by
X
(g) = −dH +
ni Yi = −dH + D.
i
It follows that D ∼ dH in Cl Pn . Clearly if Dα ∼ dα H and Dβ ∼ dβ H are
two divisors then Dα + Dβ ∼ (dα + dβ )H, so we see that Cl Pn ∼
= Z. Further,
Cl X is generated by the class of a hyperplane, and each class is represented by a
hypersurface of the corresponding degree.
Hartshorne gives some machinery for calculating Cl X in some other cases. We
won’t go through that here. There are, however, some nice examples of how the
divisor class group can be used to produce interesting geometric results. The next
example is 6.6.3 in Hartshorne.
Example 3.9. Let Q ⊂ P3 be the quadric surface defined by Z0 Z3 = Z1 Z2 . Then
Cl Q ∼
= Z ⊕ Z. We say a divisor D of Q is of type (a, b) if it corresponds to (a, b)
under this isomorphism.
Let C ⊂ P3 denote the twisted cubic curve, i.e. the image of the map P1 → P3
given by
[X0 , X1 ] 7→ [X03 , X02 X1 , X0 X12 , X13 ].
Then C certainly lies on Q. If Y is the quadric Z1 Z2 = Z02 then Y ∩ Q = C ∪ L
where L is a line. In P3 the surface Y is a divisor with Y ∼ 2H for H a hyperplane
(by Example 3.8). Then Y ∩ Q has type (2,2) (by Example II.6.6.2 [3]).
The line L has type (1,0) in Q so C = Y ∩ Q − L has type (2, 2) − (1, 0) = (1, 2).
Suppose now X ⊂ P3 is a surface such that X ∩ Q = C. Then the divisor X ∩ Q
would have type nC = (n, 2n) for some n > 0. But we have already seen that if X
has degree d then X ∼ dH and then X ∩ Q has type (d, d). Thus there is no such
X.
Let us now consider the other type of divisor: Cartier divisors. We’ll begin with
motivation. Let X be a smooth complex projective variety. We obtain a sheaf of
groups M∗ which is the multiplicative sheaf of meromorphic functions on X not
identically zero. We have another sheaf O∗ , of nowhere zero holomorphic functions
on X. A Cartier divisor on X is a global section of the quotient sheaf M∗ /O∗ .
Note that giving a Cartier divisor is precisely the same as giving an element of
Cl X. Indeed, if f is a global section of M∗ /O∗ , then there is an open cover Uα of
X so that the restriction fα of f to Uα satisfies
fα
∈ O∗ (Uα ∩ Uβ ).
fβ
That is, the global section f restricts to sections on the open cover whose difference
(as group elements) is in O∗ on the intersections. Let Y ⊂ X be a codimension
one subvariety with Y ∩ Uα ∩ Uβ 6= 0. Since fα /fβ is a nowhere zero holomorphic
function, we must have
fα
vY
= 0.
fβ
Thus vY (fα ) = vY (fβ ). We can therefore define the Weil divisor corresponding to
f as
X
(f ) =
vY (fα )Y.
Y
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JOHN KOPPER
Where for each Y we choose
P α such that Y ∩ Uα is nonempty.
Conversely, let D = i ni Yi . For each each i, and any point p ∈ Yi , Yi can be
given locally as the zero locus of a single holomorphic function gi ∈ O∗ (Yi ∩ U ) for
some U (this is a fact about smooth complex varieties, and more generally about
smooth varieties, and more more generally about “locally factorial” varieties). We
obtain an element of M∗ (Yi ∩ U ) defined by
Y
fU =
gini .
i
∗
∗
This gives a global section of M /O because on intersections Uα ∩ Uβ the gi
corresponding to Uα and Uβ which vanish there must be the same (and have the
same order of vanishing). That is, fα /fβ ∈ O∗ (Uα ∩ Uβ ).
The association of a Cartier divisor to a Weil divisor and vice-versa are clearly
mutual inverses, so we have in fact defined an isomorphism between Div X and the
group of global sections Γ(X, M∗ /O∗ ). This isomorphism will not hold in general,
but it will hold for smooth varieties. We generalize the definition of Cartier divisor
in the only way possible.
Definition 3.10. Let X be a variety. Denote by K the sheaf where K (U ) is the
set of rational functions on U . Let K ∗ be the sheaf of multiplicative groups of
invertible elements of K . Let O∗ be the sheaf of invertible regular functions. Then
a Cartier divisor on X is a global section of the quotient sheaf K ∗ /O∗ . A Cartier
divisor is principal if it is in the image of the map Γ(X, K ∗ ) → Γ(X, K ∗ /O∗ ).
Proposition 3.11. If X is a smooth variety then there is an isomorphism of groups
Div X → Γ(X, K ∗ /O∗ ) which sends principal Weil divisors to principal Cartier
divisors.
The proof of this is more or less the same as the one we gave for the complex case.
It’s worth mentioning that slightly more is true: the isomorphism holds whenever X
is locally factorial, i.e. when the local rings OX |p are unique factorization domains.
It is a theorem from commutative algebra that regular rings are unique factorization
domains, so smooth varieties are locally factorial.
Definition 3.12. The image of the map Γ(X, K ∗ ) → Γ(X, K ∗ /O∗ ) is clearly
a subgroup of Γ(X, K ∗ /O∗ ). We define the Cartier class group CaCl X as the
quotient of Γ(X, K ∗ /O∗ ) by this subgroup, i.e. Cartier divisors modulo principal
equivalence.
∼ Cl X when X is
We can interpret Proposition 3.11 as saying that CaCl X =
∼ Pic X for any variety. To understand this, we
smooth. It turns out that CaCl X =
need to associate to each Cartier divisor D a corresponding line bundle L (D) (the
notation O(D) is common). There is essentially one natural way to do so: let {Ui }
be an open cover of X so that the global section fi of K ∗ /O∗ generates on Ui . We
define a transition function ϕij on Ui ∩ Uj by ϕij = fj /fi , and let L (D) be the
line bundle defined by these transition functions.
Example 3.13. Let X = Pn and let D be the hyperplane that is the zero locus
of Z0 = 0. Then D is a Weil divisor, and the corresponding Cartier divisor is the
function g0 given by g0 = 1 on the open set Z0 = 1 and g0 = Z0 /Zi on the open
set Zi = 1. The corresponding line bundle then has transition functions
Z0 /Zj
Zi
ϕij =
=
.
Z0 /Zi
Zj
LINE BUNDLES AND DIVISORS IN ALGEBRAIC GEOMETRY
11
We recognize this as one of our favorite line bundles, O(1). For this reason, O(1)
is sometimes called the hyperplane bundle.
We have defined a map D 7→ L (D). We can construct its inverse: given a line
bundle F on X, we can associate a Cartier divisor div(F ). Let Ui be a covering
of X so that F (Ui ) is generated (as an O-module) by fi . Then [fi−1 ] is an element
of (K ∗ /O∗ )(Ui ), and fi and fj are related by fj−1 = (fi /fj )fi−1 on Ui ∩ Uj . Of
course, fi /fj ∈ O∗ (Ui ∩ Uj ) so we have in fact given a global section of K ∗ /O∗ .
Proposition 3.14. The maps D 7→ L (D) and F 7→ div(F ) are mutual two-sided
inverses.
Proof. First let D be a Cartier divisor on X. Let {Ui } be an open cover of X so
that fi generates (K ∗ /O∗ )(Ui ). Then L (D) is the line bundle given by transition
functions ϕij = fj /fi . Then the fi are also generators for L (D) so div(L (D)) is
generated on the open set Ui by fi .
Conversely, suppose a line bundle F is given. Let Ui be a cover of X so that
F (Ui ) ∼
= O(U ), generated by fi . Then div(F ) is given by fi on Ui where fj =
ϕij (fj ). Thus L (div(F )) = F .
Notice that if D is a global section of K ∗ /O∗ then the corresponding line bundle
L (D) is isomorphic to the trivial bundle O (indeed the transition functions are
the constant ones). In fact, the map D 7→ L (D) defines an isomorphism from
CaCl X → Pic X when we account for linear equivalence. This is not hard to check
(see [3] II.6.13).
Thus for any variety X we have CaCl X ∼
= Pic X. For smooth varieties we have
Cl X ∼
= CaCl X, hence Cl X ∼
= Pic X for smooth varieties.
Example 3.15. We know that Cl Pn = Z. Since Pn is smooth, it follows that
Pic Pn = Z. Since each O(m) is distinct, it follows that any invertible sheaf of Pn
is isomorphic to O(m) for some m.
Some closing remarks to conclude this section. The three groups we’ve described,
Cl X, CaCl X, and Pic X, play a somewhat analogous role to π1 (X) in topology.
There are of course some tools to help us compute them. For example if Z is a
proper closed subvariety of X then there is an exact sequence
Z → Cl X → Cl(X − Z) → 0.
This can be used to show that if Z is a curve of degree d in P2 then Cl(P2 − Z) =
Z/dZ ([3] II.6.5.1) or that Cl(P1 × P1 ) = Z ⊕ Z ([4] 14.2.O).
References
[1] P. Griffiths & J. Harris. Principles of Algebraic Geometry. Wiley Classics Edition, Hoboken
NJ, 1994.
[2] J. Harris. Algebraic Geometry: A First Course. Springer-Verlag, New York, 1991.
[3] R. Hartshorne. Algebraic Geometry. Springer-Verlag, New York, 1977.
[4] R. Vakil. Foundations of Algebraic Geometry. April 2015 Version,
https://math216.wordpress.com/.