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 RG = {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 −1 OY (U) → OX (π (U)) induces an isomorphism

−1 G OY (U) ' OX (π (U)) .

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 G a)R is a finely generated C-algebra. b) The morphism π : X −→ Spec(RG ) induced by RG ⊆ 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 = ∪αVα such that −1 −1 π|π (Vα) : π (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

∗ 2 Example: Let C act on C \{0} by scalar multiplication, where 2 2 C = Spec(C[x0, x1]). Then C \{0} = U0 ∪ U1, where

2 ±1 U0 = C \V (x0) = Spec(C[x0 , x1])

2 ±1 U1 = C \V (x1) = Spec(C[x0, x1 ]) 2 ±1 ±1 U0 ∩ U1 = C \V (x0x1) = Spec(C[x0 , x1 ]) The rings of invariants are

∗ ±1 C C[x0 , x1] = C[x1/x0]

∗ ±1 C C[x0, x1 ] = C[x0/x1] ∗ ±1 ±1 C ±1 C[x0 , x1 ] = C[(x1/x0) ]

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 1 ∗ 2 create P . Since C -orbits are closed in C \{0}, it follows that

1 2 ∗ P = (C \{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

r XΣ ' (C \Z)//G

r r for an appropriate of affine space C , exceptional set Z ⊆ C , 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 (X ) = Div (XΣ)/ . Σ Div0 (XΣ)

Remark: By the above definition, we have the following short exact sequence of affine algebraic group

∗ Σ(1) 1 −→ G −→ (C ) −→ 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

∗ Σ(1) Y hm,uρi G = {(tρ) ∈ (C ) | tρ = 1 for all m ∈ M} ρ

∗ Σ(1) Y hei ,uρi = {(tρ) ∈ (C ) | 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

n Example: The ray generators of the fan for P are n X u0 = − ei , u1 = e1, ..., un = en. i=1

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

hm,−e1−...−eni hm,e1i hm,eni t0 t1 ...tn = 1

n for all m ∈ M = Z . Taking m equal to e1, ..., en, we see that G is defined by −1 −1 t0 t1 = ... = t0 tn. Thus ∗ ∗ G = {(λ, ..., λ)|λ ∈ C }' C , ∗ n+1 which is the action of C on C 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

1 1 Example: The fan for P × P has ray generators 2 u1 = e1, u2 = −e1, u3 = e2, u4 = −e2 in N = Z . By this lemma, ∗ 4 (t1, t2, t3, t4) ∈ (C ) lies in G if and only if

hm,e1i hm,−e1i hm,e2i hm,−e2i t1 t2 t3 t4 = 1

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

−1 −1 t1t2 = t3t4 = 1.

Thus ∗ ∗ 2 G = {(µ, µ, λ, λ)|µ, λ ∈ C }' (C ) .

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

σˆ Y x = xρ. ρ/∈σ(1)

b) B(Σ) := hxσˆ|σ ∈ Σi ⊆ S is called irrelevant ideal. Remark: Σ(1) a) Spec(S) = C . b) xτˆ is the multiple of xσˆ 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) σˆ B(Σ) = hx |σ ∈ Σmax i.

where Σmax is the set of maximal cones of Σ.

Now define Σ(1) Z(Σ) = V (B(Σ)) ⊆ C . n Example: The fan for P consists of cones generated by proper Pn subsets of {u0, ..., un}, where u0 = − i=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 σˆ are σi = Cone(u0, ..., uˆi , ..., un). Then x i = xi , so that B(Σ) = hx0, ..., xni. 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 P, there is a σ ∈ Σ with Q ⊆ σ(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

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).

n Example: The fan for P consists of cones generated by proper Pn subsets of {u0, ..., un}, where u0 = − i=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

1 1 Example: The fan for P × P 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 x2x4 and the others give x1x4, x1x3, x2x3. Thus B(Σ) = hx2x4, x1x4, x1x3, x2x3i. We can check that

2 2 Z(Σ) = {0} × C × C × {0}.

* The only primitive collections are {ρ1, ρ2} and {ρ3, ρ4}, so that

2 2 Z(Σ) = V (x1, x2) ∪ V (x3, x4) = {0} × C × C × {0} by the proposition, where B(Σ) = hx1, x2i ∩ hx3, x4i.

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

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

Σ(1) σ˜ = Cone(eρ|ρ ∈ σ(1)) ⊆ R .

These cones and their faces form a fan Σe = {σ˜|σ ∈ Σ} in 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 Let Σe be the fan as above. Σ(1) a) C \Z(Σ) is the toric variety of the fan Σe. Σ(1) b) The map eρ 7→ uρ defines a map of lattices Z → N that is compatible with the fans Σe and Σ in NR. c) The resulting toric morphism

Σ(1) π : C \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 Σ(1) toric morphism π : C \Z(Σ) −→ XΣ from the above proposition. Then: a) π is a good categorical quotient for the action of G on Σ(1) C \Z(Σ), so that

Σ(1) XΣ ' (C \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:

Σ(1) XΣ ' (C \Z(Σ))//G ↑ ↑ ∗ Σ(1) TN ' (C ) /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 n of P is n n+1 ∗ P = (C \{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 1 1 representation P × P is

1 1 4 2 2 ∗ 2 C × C = (C \({0} × C ∪ C × {0}))/(C ) ,

∗ 2 where (C ) 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:

Σ(1) 0 −→ M −→ Z −→ Cl(XΣ) −→ 0

Σ(1) where α = (aρ) ∈ Z maps to the class [ΣρaρDρ] ∈ Cl (XΣ). α Q aρ Given x = ρ xρ ∈ S, we define its degree:

α deg (x ) = [ΣρaρDρ] ∈ 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) β ∗ Σ(1) gives the character χ : G −→ C . The action of G on C induces an action on S with the following property: For given f ∈ S

β −1
f ∈ Sβ ⇐⇒ g.f = χ g f for all g ∈ G β Σ(1) ⇐⇒ f (g.x) = χ (g) f (x) for all g ∈ G, x ∈ C .

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: n The total coordinate ring of P is C[x0, ..., xn]. n+1 The map Z → 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: n m n m The fan for P × P is the product of the fans of P and P . The class group is

n m n m 2 Cl(P × P ) ' Cl(P ) × Cl(P ) ' Z .

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 k V (I ) = ∅ in XΣ ⇐⇒ B(Σ) ⊆ 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 X } ←→ Σ 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

 −1 V (I ) = p ∈ XΣ| there is a x ∈ π (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) tk xm is called monomial, where m ∈ k∆. b) The monomials multiply by

tk xm.tl xn := tk+l xm+n.

c) The degree of tk xm is

deg(tk xm) := k.

d) The polytope ring of ∆ is

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

∞ M S∆ = (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).

−1 We may identity Uf with the topological space Spec(S∆[f ]0) and give it the corresponding structure of an affine scheme, where g S [f −1] = { |f , g homogeneous in S and deg(g) = deg(f s )}. ∆ 0 f s ∆

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 v 0 0 a) σF := {λ(m − m )|m ∈ ∆, m ∈ 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