Toric

1 Affine Toric Varieties

1.1 Introduction to Affine Toric Varieties

1.1.1 The Torus

Definition 1.1.1 The split torus

The split torus \({\mathbb G_m}^n\) over a scheme \(S\) is the pullback of \(\operatorname{Spec}{\mathbb Z}[x_1^{\pm 1}, \dotsc , x_n^{\pm 1}]\) along the unique map \(S \to \operatorname{Spec}{\mathbb Z}\).

Lemma 1.1.2 The split torus over \(\operatorname{Spec}R\)

The split torus over \(\operatorname{Spec}R\) is isomorphic to \(\operatorname{Spec}(R[x_1^{\pm 1}, \dotsc , x_n^{\pm 1}])\).

Proof

Ask any toddler on the street.

Definition 1.1.3 Characters of a group scheme
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For a group scheme \(G\) over \(S\), the character lattice of \(G\) is

\[ X(G) := \operatorname{Hom}_{\mathsf{GrpSch}_S}(G, {\mathbb G_m}). \]

An element \(X(G)\) is (unsurprisingly) called a character.

Proposition 1.1.4 Character lattice of the torus

Characters of the torus over a field \(k\) are isomorphic to \({\mathbb Z}^n\). \(X({\mathbb G_m}^n) = {\mathbb Z}^n\).

Proof

By Propositions 1.1.2 and 0.4.16 in turn, we have

\[ X({\mathbb G_m}^n) = \operatorname{Hom}_{\mathsf{GrpSch}}({\mathbb G_m}^n, {\mathbb G_m}) = \operatorname{Hom}(k[{\mathbb Z}], k[{\mathbb Z}^n]) = \operatorname{Hom}({\mathbb Z}, {\mathbb Z}^n) = {\mathbb Z}^n. \]
Proposition 1.1.5 The image of a torus is a torus

Let \(T_1\) and \(T_2\) be split tori over a field \(k\) and let \(\Phi : T_1 \to T_2\) be a homomorphism, then \(\Phi \) factors as

\[ T_1 \xrightarrow {\Phi } T_2 = T_1 \xrightarrow {\phi } T \xrightarrow {\iota } T_2, \]

where \(T\) is a split torus, \(\iota \) is a closed subgroup embedding and \(\phi \) is an fpqc homomorphism.

Proof

Let \(M_1=X(T_1), M_2=X(T_2)\). Define \(M\) to be the image of the homomorphism \(M_2 \to M_1\) corresponding to \(\Phi \) and take \(T = D_k(M)\). The homomorphisms \(\iota ,\phi \) correspond to the canonical quotient map \(M_2 \to M\) and the canonical inclusion \(M \to M_1\) respectively. Hence \(\Phi = \iota \circ \phi \).

\(M\) is a subgroup of a finitely-generated free abelian group \(M_1\), hence itself a finitely-generated free abelian group. Thus \(T\) is a split torus.

\(\iota \) is a closed embedding since the corresponding ring map is a quotient map with kernel generated by the kernel of \(M_2 \to M_1\).

\(\phi \) is affine, hence quasi-compact. A collection of coset representatives for \(M /M_1\) gives a basis for \(k[M]\) as a \(k[M_1]\) module, hence \(\phi \) is faithfully flat.

Proposition 1.1.6 A subgroup of a torus is a torus

Let \(T\) be a split torus. If \(H \subseteq T\) is an irreducible subgroup, then \(H\) is a split torus.

Proof

Let \(M = X(T), N=X(H)\). Since \(H\) is a closed subscheme \(M \to N\) is surjective, so \(N\) is a finitely-generated abelian group. Since \(H\) is irreducible it is connected, so \(N\) is torsion-free, hence free. Thus \(H\) is a split torus.

Definition 1.1.7 The character eigenspace

For a finite dimensional representation of a torus \(T\) on \(W\), the character eigenspace of a character \(\chi \in X(T)\) is

\[ W_m = \{ w\in W : t\cdot w = \chi (t)\text{ for all } t\in T \} . \]
Proposition 1.1.8 Decomposition into character eigenspaces

The space decomposes into the direct sum of the character eigenspaces.

Proof

TODO

Definition 1.1.9
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For a group scheme \(G\), the cocharacter lattice of \(G\) is \(\operatorname{Hom}_{\mathsf{GrpSch}_S}({\mathbb G_m}, G)\). An element is called a cocharacter or one-parameter subgroup.

Definition 1.1.10 The character-cocharacter pairing

Character lattice and one-parameter subgroup pairing.

Proposition 1.1.11 Cocharacter lattice of the torus

\(N = \operatorname{Hom}(M, {\mathbb Z}) \cong {\mathbb Z}^n\). For \(u \in N\) we write \(\lambda ^u\) for the corresponding cocharacter.

Proof

By Propositions 1.1.2 and 0.4.16 in turn, we have

\[ \mathrm{cochar}({\mathbb G_m}^n) = \operatorname{Hom}_{\mathsf{GrpSch}}({\mathbb G_m}, {\mathbb G_m}^n) = \operatorname{Hom}(k[{\mathbb Z}^n], k[{\mathbb Z}]) = \operatorname{Hom}({\mathbb Z}^n, {\mathbb Z}) \cong {\mathbb Z}^n. \]

1.1.2 The Definition of Affine Toric Variety

Definition 1.1.12
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A toric variety is a variety \(X\) with

  • an open embedding \(T := ({\mathbb C}^\times )^n \hookrightarrow X\) with dense image

  • such that the natural action \(T \times T \to T\) of the torus on itself extends to an (algebraic) action \(T \times X \to X\).

1.1.3 Lattice Points

Definition 1.1.13

Given a finite set \({\mathcal A}= \{ a_1, \dotsc , a_s\} \subseteq M\), define \(\Phi _{\mathcal A}: T \to \mathbb {A}^s\) given by \(\Phi _{\mathcal A} (t) = (\chi ^{a_1} (t), \dotsc , \chi ^{a_s} (t))\).

Definition 1.1.14

\(Y_{\mathcal A}\) is the (Zariski) closure of \(\operatorname{im}\Phi _{\mathcal A}\) in \(\mathbb A^s\).

Proposition 1.1.15

Proposition 1.1.8

Proof

TODO

1.1.4 Toric Ideals

Proposition 1.1.16

The ideal of the affine toric variety \(Y_{\mathcal A}\) is

\[ I(Y_{\mathcal A}) = \langle x^{\ell _+} - x^{\ell _-} | \ell \in L\rangle \]
Proof

See [ 1 ] .

Definition 1.1.17
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The ideal \(I_L = \langle x^\alpha - x^\beta | \alpha , \beta \in {\mathbb N}^s \text{ and } \alpha - \beta \in L\rangle \) is called the lattice ideal of the lattice \(L \subseteq {\mathbb Z}^s\).

A toric ideal is a prime lattice ideal.

Definition 1.1.18
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A toric ideal is a prime lattice ideal.

Proposition 1.1.11: an ideal is toric if and only if it’s prime and generated by binomials \(x^\alpha - x^\beta \).

Proof
Proposition 1.1.20 The spectrum of an affine monoid algebra is an affine toric variety

If \(S\) is an affine monoid, then \(\operatorname{Spec}(\Bbbk [S])\) is an affine toric variety.

Proof

Identify the torus with \(\Bbbk [x_1^{\pm 1}, \dotsc , x_n^{\pm 1}]\) using Lemma 1.1.2. \(i\) induces a morphism \(T \to \operatorname{Spec}(\Bbbk [S])\). It’s an open embedding as \(i\) gives the localization of \(\Bbbk [S]\) at \(\chi ^{a_i}\), so \(\operatorname{im}i\) is an affine open. It’s dominant as \(\operatorname{Spec}(\Bbbk [S])\) is integral and so is irreducible, and \(\operatorname{im}i\) is open and nonempty, so dense. The torus action is given by the natural restriction of comultiplication on \(\Bbbk [x_1^{\pm 1}, \dotsc , x_n^{\pm 1}]\) using Proposition 0.3.8.

Proposition 1.1.21 The character lattice of the spectrum of an affine monoid algebra

If \(S\) is an affine monoid, then the character lattice of \(\operatorname{Spec}(\Bbbk [S])\) is \({\mathbb Z}S\).

Proof

It is what it is.

If \(S\) is an affine monoid and \(\mathcal A\) is a finite set generating \(S\) as a monoid, then \(\operatorname{Spec}(\Bbbk [S]) = Y_{\mathcal A}\).

Proof

We get a \(\Bbbk \)-algebra homomorphism \(\pi : \Bbbk [x_1, \dotsc , x_s] \to \Bbbk [{\mathbb Z}S]\) given by \(\mathcal A\); this induces a morphism \(\Phi _{\mathcal A} : T \to \Bbbk ^s\). The kernel of \(\pi \) is the toric ideal of \(Y_{\mathcal A}\) and \(\pi \) is clearly surjective, so \(Y_{\mathcal A} = \mathbb V(\ker (\pi )) = \operatorname{Spec}(\Bbbk [x_1, \dotsc , x_s] / \ker (\pi )) = \operatorname{Spec}({\mathbb C}[S])\).

Definition 1.1.23

Torus action on semigroup algebra

1.1.5 Equivalence of Constructions

Proof

TFAE:

  1. \(V\) is an affine toric variety.

  2. \(V = Y_{\mathcal A}\) for some finite \(\mathcal A\).

  3. \(V\) is an affine variety defined by a toric ideal.

  4. \(V = \operatorname{Spec}\Bbbk [S]\) for an affine monoid \(S\).

Proof

1.2 Cones and Affine Toric Varieties

1.2.1 Convex Polyhedral Cones

Fix a pair of dual real vector spaces \(M\) and \(N\).

Definition 1.2.1 Convex cone generated by a set
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For a set \(S \subseteq N\), the cone generated by \(S\), aka cone hull of \(S\), is

\[ \operatorname{Cone}(S) := \left\{ \sum _{u \in S} \lambda _u u | \lambda _u \ge 0\right\} \]
Definition 1.2.2 Convex polyhedral cone
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A polyhedral cone is a set that can be written as \(\operatorname{Cone}(S)\) for some finite set \(S\).

Definition 1.2.3 Convex hull
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For a set \(S \subseteq N\), the convex hull of \(S\) is

\[ \operatorname{Conv}(S) := \left\{ \sum _{u \in S} \lambda _u | \lambda _u \ge 0, \sum _u \lambda _u = 1\right\} \]
Definition 1.2.4 Polytope
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A polytope is a set that can be written as \(\operatorname{Conv}(S)\) for some finite set \(S\).

1.2.2 Dual Cones and Faces

Definition 1.2.5 Dual cone
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Given a polyhedral cone \(\sigma \subseteq N\), its dual cone is defined by

\[ \sigma ^\vee = \{ m \in M | \forall u \in \sigma , \langle m, u\rangle \ge 0\} \]

.

Proposition 1.2.6 Dual of a polyhedral cone

If \(\sigma \) is polyhedral, then its dual \(\sigma ^\vee \) is polyhedral too.

Proof

Classic. See [ 3 ] maybe.

Proposition 1.2.7 Dual cone of a sumset

If \(\sigma _1, \sigma _2\) are two cones, then

\[ (\sigma _1 + \sigma _2)^\vee = \sigma _1^\vee \cap \sigma _2^\vee . \]
Proof

Classic. See [ 3 ] maybe.

Proposition 1.2.8 Double dual of a polyhedral cone

If \(\sigma \) is polyhedral, then \(\sigma ^{\vee \vee } = \sigma \).

Proof

Classic. See [ 3 ] maybe.

Given \(m \ne 0\) in \(M\), we get the hyperplane

\[ H_m = \{ u \in N | \langle m, u\rangle = 0\} \subseteq N \]

and the closed half-space

\[ H_m^+ = \{ u \in N | \langle m, u\rangle \ge 0\} \subseteq N. \]
Definition 1.2.9 Face of a cone
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If \(\sigma \) is a cone, then a subset of \(\sigma \) is a face iff it is the intersection of \(\sigma \) with some halfspace. We write this \(\tau \preceq \sigma \). If furthermore \(\tau \ne \sigma \), we call \(\tau \) a proper face and write \(\tau \prec \sigma \).

Definition 1.2.10 Edge of a cone

A dimension 1 face of a cone is called an edge.

Definition 1.2.11 Facet of a cone

A codimension 1 face of a cone is called a facet.

Lemma 1.2.12 Face of a polyhedral cone

If \(\sigma \) is a polyhedral cone, then every face of \(\sigma \) is a polyhedral cone.

Lemma 1.2.13 Intersection of faces

If \(\sigma \) is a polyhedral cone, then the intersection of two faces of \(\sigma \) is a face of \(\sigma \).

Proof

Classic. See [ 3 ] maybe.

Lemma 1.2.14 Face of a face

A face of a face of a polyhedral cone \(\sigma \) is again a face of \(\sigma \).

Proof

Classic. See [ 3 ] maybe.

Let \(\tau \) be a face of a polyhedral cone \(\sigma \). If \(v, w \in \sigma \) and \(v + w \in \tau \), then \(v, w \in \tau \).

Proof

Classic. See [ 3 ] maybe.

Proposition 1.2.16 Dual cone of the intersection of halfspaces

If \(\sigma = H_{m_1}^+ \cap \dots \cap H_{m_s}^+\), then

\[ \sigma ^\vee = \operatorname{Cone}(m_1, \dots , m_s). \]
Proof

Classic. See [ 3 ] maybe.

Proposition 1.2.17 Facets of a full dimensional cone

If \(\sigma \) is a full dimensional cone, then facets of \(\sigma \) are of the form \(H_m \cap \sigma \).

Proof

Classic. See [ 3 ] maybe.

Proposition 1.2.18 Intersection of facets containing a face

Every proper face \(\tau \prec \sigma \) of a polyhedral cone \(\sigma \) is the intersection of the facets of \(\sigma \) containing \(\tau \).

Proof

Classic. See [ 3 ] maybe.

Definition 1.2.19 Dual face

Given a cone \(\sigma \) and a face \(\tau \preceq \sigma \), the dual face to \(\tau \) is

\[ \tau ^* := \sigma ^\vee \cap \tau ^\perp \]
Proposition 1.2.20 The dual face is a face of the dual

If \(\tau \preceq \sigma \), then \(\tau ^* \preceq \sigma ^\vee \).

Proof

Classic. See [ 3 ] maybe.

Proposition 1.2.21 The double dual of a face

If \(\tau \preceq \sigma \), then \(\tau ^{**} = \tau \).

Proof

Classic. See [ 3 ] maybe.

Proposition 1.2.22 The dual of a face is antitone

If \(\tau ' \preceq \tau \preceq \sigma \), then \(\tau ' \preceq \tau \).

Proof

Classic. See [ 3 ] maybe.

Proposition 1.2.23 The dimension of the dual of a face

If \(\tau \preceq \sigma \), then

\[ \dim \tau + \dim \tau ^* = \dim N. \]
Proof

Classic. See [ 3 ] maybe.

1.2.3 Relative Interiors

Definition 1.2.24 Relative interior
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The relative interior, aka intrinsic interior, of a cone \(\sigma \) is the interior of \(\sigma \) as a subset of its span.

Lemma 1.2.25 The relative interior in terms of the inner product

For a cone \(\sigma \),

\[ u \in \operatorname{Relint}(\sigma ) \iff \forall m \in \sigma ^\vee \setminus \sigma ^\perp , \langle m, u\rangle {\gt} 0 \]
Proof

Classic. See [ 3 ] maybe.

Lemma 1.2.26 Relative interior of a dual face

If \(\tau \preceq \sigma \) and \(m \in \sigma ^\vee \), then

\[ m \in \operatorname{Relint}(\tau ^*) \iff \tau = H_m \cap \sigma \]
Proof

Classic. See [ 3 ] maybe.

Lemma 1.2.27 Minimal face of a cone

If \(\sigma \) is a cone, then \(W := \sigma \cap (-\sigma )\) is a subspace. Furthermore, \(W = H_m \cap \sigma \) whenever \(m \in \operatorname{Relint}(\sigma ^\vee )\).

Proof

Classic. See [ 3 ] maybe.

1.2.4 Strong Convexity

Definition 1.2.28 Salient cones
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A cone \(\sigma \) is salient, aka pointed or strongly convex, if \(\sigma \cap (-\sigma ) = \{ 0\} \).

Proposition 1.2.29 Alternative definitions of salient cones

The following are equivalent

  1. \(\sigma \) is salient

  2. \(\{ 0\} \preceq \sigma \)

  3. \(\sigma \) contains no positive dimensional subspace

  4. \(\dim \sigma ^\vee = \dim N\)

Proof

Classic. See [ 3 ] maybe.

1.2.5 Separation

Let \(\sigma _1, \sigma _2\) be polyhedral cones meeting along a common face \(\tau \). Then

\[ \tau = H_m \cap \sigma _1 = H_m \cap \sigma _2 \]

for any \(m \in \operatorname{Relint}(\sigma _1^\vee \cap (-\sigma _2)^\vee )\).

Proof

See [ 1 ] .

1.2.6 Rational Polyhedral Cones

Let \(M\) and \(N\) be dual lattices with associated vector spaces \(M_{\mathbb R}:= M \otimes _{\mathbb Z}{\mathbb R}, N_{\mathbb R}:= N \otimes _{\mathbb Z}{\mathbb R}\).

Definition 1.2.31 Rational cone

A cone \(\sigma \subseteq N_{\mathbb R}\) is rational if \(\sigma = \operatorname{Cone}(S)\) for some finite set \(S \subseteq N\).

Lemma 1.2.32 Faces of a rational cone

If \(\tau \preceq \sigma \) is a face of a rational cone, then \(\tau \) itself is rational.

Proof

Classic. See [ 3 ] maybe.

Lemma 1.2.33 The dual of a rational cone

\(\sigma ^\vee \) is a rational cone iff \(\sigma \) is.

Proof

Classic. See [ 3 ] maybe.

Definition 1.2.34 Ray generator

If \(\rho \) is an edge of a rational cone \(\sigma \), then the monoid \(\rho \cap N\) is generated by a unique element \(u_\rho \in \rho \cap N\), which we call the ray generator of \(\rho \).

Definition 1.2.35 Minimal generators

The minimal generators of a rational cone \(\sigma \) are the ray generators of its edges.

Lemma 1.2.36 A rational cone is generated by its minimal generators

A salient convex rational polyhedral cone is generated by its minimal generators.

Proof

Classic. See [ 3 ] maybe.

Definition 1.2.37 Regular cone

A salient rational polyhedral cone \(\sigma \) is regular, aka smooth, if its minimal generators form part of a \({\mathbb Z}\)-basis of \(N\).

Definition 1.2.38 Simplicial cone

A salient rational polyhedral cone \(\sigma \) is simplicial if its minimal generators are \({\mathbb R}\)-linearly independent.

1.2.7 Semigroup Algebras and Affine Toric Varieties

Definition 1.2.39 Dual lattice of a cone

If \(\sigma \subseteq N_{\mathbb R}\) is a polyhedral cone, then the lattice points

\[ S_\sigma := \sigma ^\vee \cap M \]

form a monoid.

Proposition 1.2.40 Gordan’s lemma

\(S_\sigma \) is finitely generated as a monoid.

Proof

See [ 1 ] .

Definition 1.2.41 Affine toric variety of a rational polyhedral cone

\(U_\sigma := \operatorname{Spec}{\mathbb C}[S_\sigma ]\) is an affine toric variety.

Theorem 1.2.42 Dimension of the affine toric variety of a rational polyhedral cone
\[ \dim U_\sigma = \dim N \iff \text{ the torus of $U_\sigma $ is } T_N = N \otimes _[{\mathbb Z}] {\mathbb C}^* \iff \sigma \text{ is salient}. \]
Proof

See [ 1 ] .

Proposition 1.2.43 The irreducible elements of the dual lattice of a cone

If \(\sigma \subseteq N_{\mathbb R}\) is salient of maximal dimension, then the irreducible elements of \(S_\sigma \) are precisely the minimal generators of \(\sigma ^\vee \).

Proof

See [ 1 ] .