Homogeneous Coordinate Ring - TU Kaiserslauternboehm/lehre/13_ST...The Total Coordinate Ring Toric...
Transcript of Homogeneous Coordinate Ring - TU Kaiserslauternboehm/lehre/13_ST...The Total Coordinate Ring Toric...
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
Quotients in Algebraic GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric 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 GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric 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 GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric Varieties via Polytopes
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.
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
Quotients in Algebraic GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric Varieties via Polytopes
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}.
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
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The Total Coordinate RingToric Varieties via Polytopes
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 .
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
Quotients in Algebraic GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric 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 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= ∅.
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
Quotients in Algebraic GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric Varieties via Polytopes
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.
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
Quotients in Algebraic GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric 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 GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric Varieties via Polytopes
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 .
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
Quotients in Algebraic GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric Varieties via Polytopes
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 .
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
Quotients in Algebraic GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric Varieties via Polytopes
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.
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
Quotients in Algebraic GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric Varieties via Polytopes
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.
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
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The Total Coordinate RingToric Varieties via Polytopes
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]
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
Quotients in Algebraic GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric Varieties via Polytopes
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.
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
Quotients in Algebraic GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric 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 GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric Varieties via Polytopes
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 .
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
Quotients in Algebraic GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric Varieties via Polytopes
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.
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
Quotients in Algebraic GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric 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 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}.
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
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The Total Coordinate RingToric Varieties via Polytopes
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.
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
Quotients in Algebraic GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric Varieties via Polytopes
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.
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
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The Total Coordinate RingToric 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 GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric 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
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.
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
Quotients in Algebraic GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric Varieties via Polytopes
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}.
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
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The Total Coordinate RingToric 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 withQ ⊆ σ(1).
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
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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}.
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
Quotients in Algebraic GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric 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 . 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〉.
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
Quotients in Algebraic GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric Varieties via Polytopes
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.
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
Quotients in Algebraic GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric Varieties via Polytopes
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.
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
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The Total Coordinate RingToric Varieties via Polytopes
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.
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
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We have a commutative diagram:
XΣ ' (CΣ(1)\Z (Σ))//G↑ ↑
TN ' (C∗)Σ(1)/G
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
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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.
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
Quotients in Algebraic GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric Varieties via Polytopes
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.
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
Quotients in Algebraic GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric 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 GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric Varieties via Polytopes
In this section we will explore how this ring relates to the algebraand geometry of XΣ.
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
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The Total Coordinate RingToric Varieties via Polytopes
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 .
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
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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.
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
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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.
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
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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.
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
Quotients in Algebraic GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric 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
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The Total Coordinate RingToric Varieties via Polytopes
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.
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
Quotients in Algebraic GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric Varieties via Polytopes
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
}
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
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The Total Coordinate RingToric Varieties via Polytopes
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.
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
Quotients in Algebraic GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric 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 GeometryQuotient Construction of Toric Varieties
The Total Coordinate RingToric 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 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 ∈ ∆}.
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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∆].
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
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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 .
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
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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}.
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
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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).
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
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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∆).
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
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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 ∆.
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring
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Theorem
P∆ = X (NF (∆)).
Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring