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Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Homogeneous Coordinate Ring
Students: Tien Mai Nguyen, Bin Nguyen
Kaiserslautern University
Algebraic Group
June 14, 2013
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Outline
1
Quotients in Algebraic Geometry
2
Quotient Construction of Toric Varieties
3
The Total Coordinate Ring
4
Toric Varieties via Polytopes
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Outline
1
Quotients in Algebraic Geometry
2
Quotient Construction of Toric Varieties
3
The Total Coordinate Ring
4
Toric Varieties via Polytopes
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Definition
Let G be a group acting on a variety X = Spec(R), R is a
K -algebra. Then the following map
G × R −→ R
(g , f ) 7−→ g .f
defined by (g .f )(x) = f (g −1 .x) for all x ∈ X is an action of G on
R.
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Remark:
a) The group acting as above is induced by the group acting of
G on X .
b) The above acting gives two objects, namely the set G -orbits
X /G = {G .x|x ∈ X } and the ring of invariants
R G = {f ∈ R|g .f = f , for all g ∈ G }.
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Definition
Let G act on X and let π : X −→ Y be morphism that is constant
on G -orbits. Then π is called a good categorical quotient if:
a) If U ⊂ Y is open, then the natural map
OY (U) → OX (π −1 (U)) induces an isomorphism
OY (U) ' OX (π −1 (U))G .
b) If W ⊆ X is closed and G -invariant, then π(W ) ⊆ Y is
closed.
c) If W1 , W2 are closed, disjoint, and G -invariant in X , then
π(W1 ) and π(W2 ) are disjoint in Y .
We often write a good categorical quotient as π : X −→ X //G .
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Theorem: Let π : X → X //G be a good categorical quotient.
Then:
a) Given any diagram
where φ is a morphism of varieties such that φ(g .x) = φ(x)
for g ∈ G and x ∈ X , there is unique morphism φ making the
diagram commute, i.e., φ ◦ π = φ.
b) π is surjective.
c) A subset U ⊆ X //G is open iff π −1 (U) ⊆ X is open.
d) x, y ∈ X , we have π(x) = π(y ) ⇔ G .x ∩ G .y 6= ∅.
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Definition
a) A subgroup G of GLn (C) is called an affine algebraic group if
G is a subvariety of GLn (C).
b) Let G be an affine algebraic group acting on a variety X . The
G -action is called algebraic action if the action
G × X −→ X
(g , x) 7−→ g .x
defines a morphism.
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Proposition
Let an affine algebraic group G act algebraically on a variety X ,
and assume that a good categorical quotient π : X −→ X //G .
Then:
a) If p ∈ X //G , then π −1 (p) contains a unique closed G -orbit.
b) π induces a bijection {closed G-orbits in X } ' X //G .
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Proposition
Let π : X −→ X //G be a good categorical quotient. Then the
following are equivalent:
a) All G -orbits are closed in X .
b) Given x, y ∈ X , we have
π(x) = π(y ) ⇐⇒ x and y lie in the same G-orbit.
c) π induces a bijection {G-orbits in X } ' X //G .
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Definition
A good categorical quotient is called a good geometric quotient if
it satisfies the condition of the above proposition.
We write a good geometric quotient as π : X → X /G .
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Definition
An affine algebraic group G is called reductive if its maximal
connected solvable subgroup is a torus.
Proposition
Let G be a reductive group acting algebraically on an affine variety
X = Spec(R). Then
a) R G is a finely generated C-algebra.
b) The morphism π : X −→ Spec(R G ) induced by R G ⊆ R is a
good categorical quotient.
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Proposition
Let G act on X and let π : X → Y be a morphism of varieties that
is constant on G -orbits. If Y has an open cover Y = ∪α Vα such
that
π|π−1 (Vα ) : π −1 (Vα ) −→ Vα
is a good categorical quotient for every α, then π : X −→ Y is a
good categorical quotient.
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Example: Let C∗ act on C2 \{0} by scalar multiplication, where
C2 = Spec(C[x0 , x1 ]). Then C2 \{0} = U0 ∪ U1 , where
U0 = C2 \V (x0 ) = Spec(C[x0±1 , x1 ])
U1 = C2 \V (x1 ) = Spec(C[x0 , x1±1 ])
U0 ∩ U1 = C2 \V (x0 x1 ) = Spec(C[x0±1 , x1±1 ])
The rings of invariants are
∗
C[x0±1 , x1 ]C = C[x1 /x0 ]
∗
C[x0 , x1±1 ]C = C[x0 /x1 ]
∗
C[x0±1 , x1±1 ]C = C[(x1 /x0 )±1 ]
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
It follows that the Vi = Ui //C∗ glue together in the usual way to
create P1 . Since C∗ -orbits are closed in C2 \{0}, it follows that
P1 = (C2 \{0})/C∗
is a good geometric quotient.
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Outline
1
Quotients in Algebraic Geometry
2
Quotient Construction of Toric Varieties
3
The Total Coordinate Ring
4
Toric Varieties via Polytopes
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Let XΣ be the toric variety of a fan Σ in NR . The goal is to
construct XΣ as a good categorical quotient
XΣ ' (Cr \Z )//G
for an appropriate of affine space Cr , exceptional set Z ⊆ Cr , and
reductive group G .
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Definition
Let XΣ be the toric variety of fan Σ in N(R) . Assume that XΣ has
no torus factor. We define
G = HomZ (Cl(XΣ ), C∗ )
where Cl (XΣ ) = Div (XΣ )/Div0 (XΣ ).
Remark: By the above definition, we have the following short
exact sequence of affine algebraic group
1 −→ G −→ (C∗ )Σ(1) −→ TN −→ 1.
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Lemma
Let G be as in the above definition. Then:
a) Cl(XΣ ) is the character group of G .
b) G is isomorphic to a product of a torus and a finite Abelian
group. In particular, G is reductive.
c) Give a basis e1 , ..., en of M. We have
Y hm,uρ i
G = {(tρ ) ∈ (C∗ )Σ(1) |
tρ
= 1 for all m ∈ M}
ρ
∗ Σ(1)
= {(tρ ) ∈ (C )
|
Y
hei ,uρ i
tρ
= 1 for 1 ≤ i ≤ n}.
ρ
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Example: The ray generators of the fan for Pn are
u0 = −
n
X
ei , u1 = e1 , ..., un = en .
i=1
By the above lemma, (t0 , ..., tn ) ∈ (C∗ )n+1 lies in G if and only if
hm,−e1 −...−en i hm,e1 i
hm,e i
t1
...tn n
t0
=1
for all m ∈ M = Zn . Taking m equal to e1 , ..., en , we see that G is
defined by
t0−1 t1 = ... = t0−1 tn .
Thus
G = {(λ, ..., λ)|λ ∈ C∗ } ' C∗ ,
which is the action of C∗ on Cn+1 given by scalar multiplication.
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Example: The fan for P1 × P1 has ray generators
u1 = e1 , u2 = −e1 , u3 = e2 , u4 = −e2 in N = Z2 . By this lemma,
(t1 , t2 , t3 , t4 ) ∈ (C∗ )4 lies in G if and only if
hm,e1 i hm,−e1 i hm,e2 i hm,−e2 i
t2
t3
t4
t1
=1
for all m ∈ M = Z2 . Taking m equal to e1 , e2 , we obtain
t1 t2−1 = t3 t4−1 = 1.
Thus
G = {(µ, µ, λ, λ)|µ, λ ∈ C∗ } ' (C∗ )2 .
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Definition
Let XΣ be the toric variety of fan Σ in N(R) .
S := C[xρ |ρ ∈ Σ(1)]
is called the homogeneous coordinate ring of XΣ .
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Definition
Let XΣ be the toric variety of fan Σ in N(R) .
a) For each cone σ ∈ Σ, define the monomial
Y
x σ̂ =
xρ .
ρ∈σ(1)
/
b)
B(Σ) := hx σ̂ |σ ∈ Σi ⊆ S
is called irrelevant ideal.
Remark:
a) Spec(S) = CΣ(1) .
b) x τ̂ is the multiple of x σ̂ whenever τ is a face of σ.
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
c)
B(Σ) = hx σ̂ |σ ∈ Σmax i.
where Σmax is the set of maximal cones of Σ.
Now define
Z (Σ) = V (B(Σ)) ⊆ CΣ(1) .
Example: The fan for Pn consists of P
cones generated by proper
subsets of {u0 , ..., un }, where u0 = − ni=1 ei , u1 = e1 , ..., un = en .
Let ui generate ρi for 0 ≤ i ≤ n and xi be the corresponding
variable in the total coordinate ring. The maximal cones of the fan
are σi = Cone(u0 , ..., ûi , ..., un ). Then x σ̂i = xi , so that
B(Σ) = hx0 , ..., xn i. Hence Z (Σ) = {0}.
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Definition
A subset P ⊆ Σ(1) is a primitive collection if:
a) P * σ(1) for all σ ∈ Σ.
b) For every proper subset Q
Q ⊆ σ(1).
Tien Mai Nguyen, Bin Nguyen
P, there is a σ ∈ Σ with
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Proposition
The Z (Σ) as a union of irreducible components is given by
[
Z (Σ) =
V (xρ |ρ ∈ P),
P
where the union is over all primitive collections P ⊆ Σ(1).
Example: The fan for Pn consists of P
cones generated by proper
subsets of {u0 , ..., un }, where u0 = − ni=1 ei , u1 = e1 , ..., un = en .
The only primitive collection is {ρ0 , ..., ρn }, so
Z (Σ) = V (x0 , ..., xn ) = {0}.
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Example: The fan for P1 × P1 has ray generators u= e1 , u2 = −e1 ,
u3 = e2 , u4 = −e2 . each ui gives a ray ρi and a variable xi . We
compute Z (Σ) in two ways:
* The maximal cone Cone(u1 , u3 ) gives the monomial x2 x4 and the
others give x1 x4 , x1 x3 , x2 x3 . Thus B(Σ) = hx2 x4 , x1 x4 , x1 x3 , x2 x3 i.
We can check that
Z (Σ) = {0} × C2 × C2 × {0}.
* The only primitive collections are {ρ1 , ρ2 } and {ρ3 , ρ4 }, so that
Z (Σ) = V (x1 , x2 ) ∪ V (x3 , x4 ) = {0} × C2 × C2 × {0}
by the proposition, where B(Σ) = hx1 , x2 i ∩ hx3 , x4 i.
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Let {eρ |ρ ∈ Σ(1)} be the standard basis of the lattice ZΣ(1) . For
each σ ∈ Σ, define the cone
σ̃ = Cone(eρ |ρ ∈ σ(1)) ⊆ RΣ(1) .
e = {σ̃|σ ∈ Σ} in
These cones and their faces form a fan Σ
Σ(1)
Σ(1)
(Z
)R = R
. This fan has the following properties.
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Proposition
e be the fan as above.
Let Σ
e
a) CΣ(1) \Z (Σ) is the toric variety of the fan Σ.
b) The map eρ 7→ uρ defines a map of lattices ZΣ(1) → N that is
e and Σ in NR .
compatible with the fans Σ
c) The resulting toric morphism
π : CΣ(1) \Z (Σ) −→ XΣ
is constant on G -orbits.
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Theorem
Let XΣ be the toric variety without torus factors and consider the
toric morphism π : CΣ(1) \Z (Σ) −→ XΣ from the above
proposition. Then:
a) π is a good categorical quotient for the action of G on
CΣ(1) \Z (Σ), so that
XΣ ' (CΣ(1) \Z (Σ))//G .
b) π is a good geometric quotient if and only if Σ is simplicial.
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
We have a commutative diagram:
XΣ ' (CΣ(1) \Z (Σ))//G
↑
↑
TN '
(C∗ )Σ(1) /G
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Example: From some examples above, the quotient representation
of Pn is
Pn = (Cn+1 \{0})/C∗ ,
where C∗ acts by scalar multiplication.
This is a good geometric quotient since Σ is a smooth and hence
simplicial.
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Example: Also from some example before, the quotient
representation P1 × P1 is
C1 × C1 = (C4 \({0} × C2 ∪ C2 × {0}))/(C∗ )2 ,
where (C∗ )2 acts via (µ, λ).(a, b, c, d) = (µa, µb, λa, λd).
This is again a good geometric quotient.
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Outline
1
Quotients in Algebraic Geometry
2
Quotient Construction of Toric Varieties
3
The Total Coordinate Ring
4
Toric Varieties via Polytopes
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
In this section we will explore how this ring relates to the algebra
and geometry of XΣ .
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Definition
Let XΣ be a toric variety without torus factor. Its total coordinate
ring is
S = C[xρ |ρ ∈ Σ(1)].
We have the sequence:
0 −→ M −→ ZΣ(1) −→ Cl(XΣ ) −→ 0
Σ(1) maps to the class [Σ a D ] ∈ Cl (X ).
where α = (a
ρ) ∈ Z
ρ ρ ρ
Σ
Q
a
Given x α = ρ xρ ρ ∈ S, we define its degree:
deg (x α ) = [Σρ aρ Dρ ] ∈ Cl (XΣ ) .
For β ∈ Cl (XΣ ), Sβ denotes the corresponding grade piece of S.
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Remark: The grading on S is closely related to
G = HomZ (Cl (XΣ ) , C∗ ) .
Cl (XΣ ) is the character group of G, where as usual β ∈ Cl (XΣ )
gives the character χβ : G −→ C∗ . The action of G on CΣ(1)
induces an action on S with the following property: For given
f ∈S
f ∈ Sβ ⇐⇒ g .f = χβ g −1 f for all g ∈ G
⇐⇒ f (g .x) = χβ (g ) f (x) for all g ∈ G , x ∈ CΣ(1) .
We say that f ∈ Sβ is homogeneous of degree β.
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Example:
The total coordinate ring of Pn is C[x0 , ..., xn ].
The map Zn+1 → Z is (a0 , ..., an ) 7→ a0 + ... + an .
This gives the grading on C[x0 , .., xn ] where each variable xi has
degree 1, so that ”homogeneous polynomial” has the usual
meaning.
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Example:
The fan for Pn × Pm is the product of the fans of Pn and Pm . The
class group is
Cl(Pn × Pm ) ' Cl(Pn ) × Cl(Pm ) ' Z2 .
The total coordinate ring is C[x0 , ..., xn , y0 , ..., ym ], where
deg (xi ) = (1, 0) deg (yi ) = (0, 1).
For this ring, ”homogeneous polynomial” means bihomogeneous
polynomial.
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Proposition
Let S be the total coordinate ring of the simplicial toric variety XΣ .
Then:
a) If I ⊆ S is a homogeneous ideal, then
V (I ) = {π(x) ∈ XΣ |f (x) = 0 for all f ∈ I }
is a closed subvarieties of XΣ .
b) All closed subvarieties of XΣ arise this way.
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Proposition (The Toric Nullstellensazt)
Let XΣ be the simplicial toric variety with total coordinate ring S
and irrelevant ideal B(Σ) ⊆ S. If I ⊆ S is a homogeneous ideal,
then
V (I ) = ∅ in XΣ ⇐⇒ B(Σ)k ⊆ I for some k ≥ 0.
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Proposition (The Toric Ideal-Variety Correspondence)
Let XΣ be a simplicial toric variety. Then there is a bijective
correspondence
radical homogeneous ideals
{closed subvarieties of XΣ } ←→
I ⊆ B(Σ) ⊆ S
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
When XΣ is not simplicial, there is still a relation between ideals in
the total coordinate ring and closed subvarieties of XΣ .
Proposition
Let S be the total coordinate ring of the toric variety XΣ . Then:
a) If I ⊆ S is a homogeneous ideal, then
V (I ) = p ∈ XΣ | there is a x ∈ π −1 (p), f (x) = 0 ∀f ∈ I
is a closed subvariety of XΣ .
b) All closed subvarieties of XΣ arise this way.
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Outline
1
Quotients in Algebraic Geometry
2
Quotient Construction of Toric Varieties
3
The Total Coordinate Ring
4
Toric Varieties via Polytopes
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Definition
a) A polytope ∆ ⊂ MR is the convex hull of affine set of points.
b) The dimension of ∆ is the dimension of the subspace spanned
by the difference {m1 − m2 |m1 , m2 ∈ ∆}.
c) ∆ is called integral if the vertice of ∆ lie in M.
d) Let ∆1 , ..., ∆k be polytopes. We define
∆1 + ... + ∆k = {m1 + ... + mk |mi ∈ ∆i ; i = 1, ..., k}.
We also denote k∆ := ∆ + ... + ∆ for k ∈ N. We have
k∆ = {km|m ∈ ∆}.
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Definition
Let ∆ be a polytope. We define
a) t k x m is called monomial, where m ∈ k∆.
b) The monomials multiply by
t k x m .t l x n := t k+l x m+n .
c) The degree of t k x m is
deg (t k x m ) := k.
d) The polytope ring of ∆ is
S∆ := C[t k x m |k ∈ N, m ∈ k∆].
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Remark:
a) The definition of the monomials multiply is well-defined.
Indeed, because m ∈ k∆, m0 ∈ l∆ we get m + m0 ∈ (k + l)∆.
b) S∆ = C[t k x m |m ∈ k∆] is a grade ring. Hence
S∆ =
∞
M
(S∆ )k .
k=0
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Definition
Let S∆ be a polytope ring of ∆.
L
+
a) S∆
:= k≥1 (S∆ )k is called irrelevant ideal.
b) T := {P|P is a homogeneous prime ideal of S∆ and P 6⊃
+
S∆
}. P ∈ T is called relevant prime ideal of S∆ .
c) For any homogeneous ideal I of S∆ , we define
Z (I ) := {P|P is a relevant prime of S∆ and P ⊃ I }.
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Proposition
a) If {Ij } is a family of homogeneous ideals in S∆ then
∩j Z (Ij ) = Z (∪j Ij ).
b) If I1 , I2 are homogeneous ideals then
Z (I1 ) ∪ Z (I2 ) = Z (I1 ∩ I2 ).
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
T is a topological space whose closed sets are Z (I ), I is a
homogeneous ideal of S∆ .
Let f be any homogeneous element of S∆ of degree 1. We set
Uf := T \Z (hf i).
We may identity Uf with the topological space Spec(S∆ [f −1 ]0 )
and give it the corresponding structure of an affine scheme, where
S∆ [f −1 ]0 = {
g
|f , g homogeneous in S∆ and deg (g ) = deg (f s )}.
fs
We will write (Proj(S∆ ))f for this open affine subscheme of
Proj(S∆ ).
Let P∆ = Proj(S∆ ).
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Definition
Let ∆ be a polytope and F be a nonempty face of ∆. We define
a) σFv := {λ(m − m0 )|m ∈ ∆, m0 ∈ F , λ ≥ 0} ⊆ MR is a cone
and its dual is a cone σF ⊂ NR .
b) NF (∆) := {σF |F is nonempty face of ∆} is normal face of ∆.
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring
Quotients in Algebraic Geometry
Quotient Construction of Toric Varieties
The Total Coordinate Ring
Toric Varieties via Polytopes
Theorem
P∆ = X (NF (∆)).
Tien Mai Nguyen, Bin Nguyen
Homogeneous Coordinate Ring