1 Affine Toric Varieties

1.1 Introduction to Affine Toric Varieties

1.1.1 The Torus

Definition 1.1.1 The split torus

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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.

Proposition 1.1.10 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

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

Definition 1.1.11 The character-cocharacter pairing

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Character lattice and one-parameter subgroup pairing.

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

$Y_{\mathcal A}$ is a toric variety.

Proof ▶

TODO

Proposition 1.1.16

The character lattice of the torus of $Y_{\mathcal A}$ is ${\mathbb Z}{\mathcal A}$.

Proof ▶

$\Phi _{\mathcal A}: T_N \to {\mathbb G_m}^s$ factors through the torus of $Y_{\mathcal A}$. The conclusion follows from looking at the corresponding maps of character lattices.

1.1.4 Toric Ideals

Proposition 1.1.17

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 1.1.18

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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 1.1.19

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A toric ideal is a prime lattice ideal.

Proposition 1.1.20

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 1.1.21 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 0.3.16.

Proposition 1.1.22

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 $Y_{\mathcal A} = \mathbb V(\ker (\pi )) = \operatorname{Spec}(\Bbbk [x_1, \dotsc , x_s] / \ker (\pi )) = \operatorname{Spec}({\mathbb C}[S])$.

Proposition 1.1.23 The character lattice of the spectrum of an affine monoid algebra

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

Proof ▶

It is what it is.

1.1.5 Equivalence of Constructions

Definition 1.1.24

There is a torus action on the semigroup algebra ${\mathbb C}[M]$: given $t\in T_N$ and $f\in {\mathbb C}[M]$ define

\[ t \cdot f = (p \mapsto f(t^{-1}p)). \]

Lemma 1.1.25

Let $A \subseteq {\mathbb C}[M]$ be a stable subspace, then

\[ A = \bigoplus _{\chi ^m \in A} {\mathbb C}\cdot \chi ^m. \]

Proof ▶

TODO

Theorem 1.1.26

TFAE:

$V$ is an affine toric variety.
$V = Y_{\mathcal A}$ for some finite $\mathcal A$.
$V$ is an affine variety defined by a toric ideal.
$V = \operatorname{Spec}\Bbbk [S]$ for an affine monoid $S$.

Proof ▶

TODO

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

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

Proof ▶

Classic, use Fourier-Motzkin eliminiation.

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.

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.

Lemma 1.2.15

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.

Proof ▶

Classic. See [ 3 ] maybe.

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

$\sigma $ is salient
$\{ 0\} \preceq \sigma $
$\sigma $ contains no positive dimensional subspace
$\dim \sigma ^\vee = \dim N$

Proof ▶

Classic. See [ 3 ] maybe.

1.2.5 Separation

Lemma 1.2.30 Separation lemma

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 ] .