Homogeneous Coordinate Ring - TU Kaiserslauternboehm/lehre/13_ST...The Total Coordinate Ring Toric Varieties via Polytopes Homogeneous Coordinate Ring Students: Tien Mai Nguyen, Bin

Quotients in Algebraic GeometryQuotient Construction of Toric Varieties

The Total Coordinate RingToric Varieties via Polytopes

Homogeneous Coordinate Ring

Students: Tien Mai Nguyen, Bin Nguyen

Kaiserslautern UniversityAlgebraic Group

June 14, 2013

Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring



Outline

1 Quotients in Algebraic Geometry

2 Quotient Construction of Toric Varieties

3 The Total Coordinate Ring

4 Toric Varieties via Polytopes




Outline








Definition

Let G be a group acting on a variety X = Spec(R), R is aK -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 onR.




Remark:

a) The group acting as above is induced by the group acting ofG on X .

b) The above acting gives two objects, namely the set G -orbitsX/G = {G .x |x ∈ X} and the ring of invariantsRG = {f ∈ R|g .f = f , for all g ∈ G}.




Definition

Let G act on X and let π : X −→ Y be morphism that is constanton G -orbits. Then π is called a good categorical quotient if:

a) If U ⊂ Y is open, then the natural mapOY (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 isclosed.

c) If W1,W2 are closed, disjoint, and G -invariant in X , thenπ(W1) and π(W2) are disjoint in Y .

We often write a good categorical quotient as π : X −→ X//G .




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 thediagram 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= ∅.




Definition

a) A subgroup G of GLn(C) is called an affine algebraic group ifG is a subvariety of GLn(C).

b) Let G be an affine algebraic group acting on a variety X . TheG -action is called algebraic action if the action

G × X −→ X

(g , x) 7−→ g .x

defines a morphism.




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 .




Proposition

Let π : X −→ X//G be a good categorical quotient. Then thefollowing 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 .




Definition

A good categorical quotient is called a good geometric quotient ifit satisfies the condition of the above proposition.

We write a good geometric quotient as π : X → X/G .




Definition

An affine algebraic group G is called reductive if its maximalconnected solvable subgroup is a torus.

Proposition

Let G be a reductive group acting algebraically on an affine varietyX = Spec(R). Then

a) RG is a finely generated C-algebra.

b) The morphism π : X −→ Spec(RG ) induced by RG ⊆ R is agood categorical quotient.




Proposition

Let G act on X and let π : X → Y be a morphism of varieties thatis constant on G -orbits. If Y has an open cover Y = ∪αVα suchthat

π|π−1(Vα) : π−1(Vα) −→ Vα

is a good categorical quotient for every α, then π : X −→ Y is agood categorical quotient.




Example: Let C∗ act on C2\{0} by scalar multiplication, whereC2 = Spec(C[x0, x1]). Then C2\{0} = U0 ∪ U1, where

U0 = C2\V (x0) = Spec(C[x±10 , x1])

U1 = C2\V (x1) = Spec(C[x0, x±11 ])

U0 ∩ U1 = C2\V (x0x1) = Spec(C[x±10 , x±1

1 ])

The rings of invariants are

C[x±10 , x1]C

∗= C[x1/x0]

C[x0, x±11 ]C

∗= C[x0/x1]

C[x±10 , x±1

1 ]C∗

= C[(x1/x0)±1]




It follows that the Vi = Ui//C∗ glue together in the usual way tocreate P1. Since C∗-orbits are closed in C2\{0}, it follows that

P1 = (C2\{0})/C∗

is a good geometric quotient.




Outline








Let XΣ be the toric variety of a fan Σ in NR. The goal is toconstruct XΣ as a good categorical quotient

XΣ ' (Cr\Z )//G

for an appropriate of affine space Cr , exceptional set Z ⊆ Cr , andreductive group G .




Definition

Let XΣ be the toric variety of fan Σ in N(R). Assume that XΣ hasno 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 shortexact sequence of affine algebraic group

1 −→ G −→ (C∗)Σ(1) −→ TN −→ 1.




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 Abeliangroup. In particular, G is reductive.

c) Give a basis e1, ..., en of M. We have

G = {(tρ) ∈ (C∗)Σ(1)|∏ρ

t〈m,uρ〉ρ = 1 for all m ∈ M}

= {(tρ) ∈ (C∗)Σ(1)|∏ρ

t〈ei ,uρ〉ρ = 1 for 1 ≤ i ≤ n}.




Example: The ray generators of the fan for Pn are

u0 = −n∑

i=1

ei , u1 = e1, ..., un = en.

By the above lemma, (t0, ..., tn) ∈ (C∗)n+1 lies in G if and only if

t〈m,−e1−...−en〉0 t

〈m,e1〉1 ...t

〈m,en〉n = 1

for all m ∈ M = Zn. Taking m equal to e1, ..., en, we see that G isdefined by

t−10 t1 = ... = t−1

0 tn.

ThusG = {(λ, ..., λ)|λ ∈ C∗} ' C∗,

which is the action of C∗ on Cn+1 given by scalar multiplication.




Example: The fan for P1 × P1 has ray generatorsu1 = 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

t〈m,e1〉1 t

〈m,−e1〉2 t

〈m,e2〉3 t

〈m,−e2〉4 = 1

for all m ∈ M = Z2. Taking m equal to e1, e2, we obtain

t1t−12 = t3t−1

4 = 1.

Thus

G = {(µ, µ, λ, λ)|µ, λ ∈ C∗} ' (C∗)2.




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.




Definition

Let XΣ be the toric variety of fan Σ in N(R).

a) For each cone σ ∈ Σ, define the monomial

x σ =∏

ρ/∈σ(1)

xρ.

b)B(Σ) := 〈x σ|σ ∈ Σ〉 ⊆ S

is called irrelevant ideal.

Remark:

a) Spec(S) = CΣ(1).

b) x τ is the multiple of x σ whenever τ is a face of σ.




c)

B(Σ) = 〈x σ|σ ∈ Σmax〉.

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 cones generated by propersubsets of {u0, ..., un}, where u0 = −

∑ni=1 ei , u1 = e1, ..., un = en.

Let ui generate ρi for 0 ≤ i ≤ n and xi be the correspondingvariable in the total coordinate ring. The maximal cones of the fanare σi = Cone(u0, ..., ui , ..., un). Then x σi = xi , so thatB(Σ) = 〈x0, ..., xn〉. Hence Z (Σ) = {0}.




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 P, there is a σ ∈ Σ withQ ⊆ σ(1).




Proposition

The Z (Σ) as a union of irreducible components is given by

Z (Σ) =⋃P

V (xρ|ρ ∈ P),

where the union is over all primitive collections P ⊆ Σ(1).

Example: The fan for Pn consists of cones generated by propersubsets 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}.




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 . Wecompute Z (Σ) in two ways:

* The maximal cone Cone(u1, u3) gives the monomial x2x4 and theothers give x1x4, x1x3, x2x3. Thus B(Σ) = 〈x2x4, x1x4, x1x3, x2x3〉.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) = 〈x1, x2〉 ∩ 〈x3, x4〉.




Let {eρ|ρ ∈ Σ(1)} be the standard basis of the lattice ZΣ(1). Foreach σ ∈ Σ, define the cone

σ = Cone(eρ|ρ ∈ σ(1)) ⊆ RΣ(1).

These cones and their faces form a fan Σ = {σ|σ ∈ Σ} in(ZΣ(1))R = RΣ(1). This fan has the following properties.




Proposition

Let Σ be the fan as above.

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

compatible with the fans Σ and Σ in NR.

c) The resulting toric morphism

π : CΣ(1)\Z (Σ) −→ XΣ

is constant on G -orbits.




Theorem

Let XΣ be the toric variety without torus factors and consider thetoric morphism π : CΣ(1)\Z (Σ) −→ XΣ from the aboveproposition. Then:

a) π is a good categorical quotient for the action of G onCΣ(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.




We have a commutative diagram:

XΣ ' (CΣ(1)\Z (Σ))//G↑ ↑

TN ' (C∗)Σ(1)/G




Example: From some examples above, the quotient representationof 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 hencesimplicial.




Example: Also from some example before, the quotientrepresentation 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.




Outline








In this section we will explore how this ring relates to the algebraand geometry of XΣ.




Definition

Let XΣ be a toric variety without torus factor. Its total coordinatering is

S = C[xρ|ρ ∈ Σ(1)].

We have the sequence:

0 −→ M −→ ZΣ(1) −→ Cl(XΣ) −→ 0

where α = (aρ) ∈ ZΣ(1) maps to the class [ΣρaρDρ] ∈ Cl (XΣ).Given xα =

∏ρ x

aρρ ∈ S , we define its degree:

deg (xα) = [ΣρaρDρ] ∈ Cl (XΣ) .

For β ∈ Cl (XΣ), Sβ denotes the corresponding grade piece of S .




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 givenf ∈ 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.




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 hasdegree 1, so that ”homogeneous polynomial” has the usualmeaning.




Example:The fan for Pn × Pm is the product of the fans of Pn and Pm. Theclass 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 bihomogeneouspolynomial.




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.




Proposition (The Toric Nullstellensazt)

Let XΣ be the simplicial toric variety with total coordinate ring Sand 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.




Proposition (The Toric Ideal-Variety Correspondence)

Let XΣ be a simplicial toric variety. Then there is a bijectivecorrespondence

{closed subvarieties of XΣ} ←→{

radical homogeneous idealsI ⊆ B(Σ) ⊆ S

}




When XΣ is not simplicial, there is still a relation between ideals inthe 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.




Outline








Definition

a) A polytope ∆ ⊂ MR is the convex hull of affine set of points.

b) The dimension of ∆ is the dimension of the subspace spannedby 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 ∈ ∆}.




Definition

Let ∆ be a polytope. We define

a) tkxm is called monomial, where m ∈ k∆.

b) The monomials multiply by

tkxm.t lxn := tk+lxm+n.

c) The degree of tkxm is

deg(tkxm) := k .

d) The polytope ring of ∆ is

S∆ := C[tkxm|k ∈ N,m ∈ k∆].




Remark:

a) The definition of the monomials multiply is well-defined.Indeed, because m ∈ k∆,m′ ∈ l∆ we get m + m′ ∈ (k + l)∆.

b) S∆ = C[tkxm|m ∈ k∆] is a grade ring. Hence

S∆ =∞⊕k=0

(S∆)k .




Definition

Let S∆ be a polytope ring of ∆.

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}.




Proposition

a) If {Ij} is a family of homogeneous ideals in S∆ then

∩jZ (Ij) = Z (∪j Ij).

b) If I1, I2 are homogeneous ideals then

Z (I1) ∪ Z (I2) = Z (I1 ∩ I2).




T is a topological space whose closed sets are Z (I ), I is ahomogeneous ideal of S∆.

Let f be any homogeneous element of S∆ of degree 1. We set

Uf := T \Z (〈f 〉).

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 s|f , g homogeneous in S∆ and deg(g) = deg(f s)}.

We will write (Proj(S∆))f for this open affine subscheme ofProj(S∆).

Let P∆ = Proj(S∆).




Definition

Let ∆ be a polytope and F be a nonempty face of ∆. We define

a) σvF := {λ(m −m′)|m ∈ ∆,m′ ∈ F , λ ≥ 0} ⊆ MR is a coneand its dual is a cone σF ⊂ NR.

b) NF (∆) := {σF |F is nonempty face of ∆} is normal face of ∆.




Theorem

P∆ = X (NF (∆)).


Documents

Homogeneous Coordinate Ring - TU Kaiserslauternboehm/lehre/13_ST...The Total Coordinate Ring Toric Varieties via Polytopes Homogeneous Coordinate Ring Students: Tien Mai Nguyen, Bin