1 Affine Toric Varieties
1.1 Introduction to Affine Toric Varieties
1.1.1 The 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 \(S \to \operatorname{Spec}{\mathbb Z}\).
The split torus over \(\operatorname{Spec}R\) is isomorphic to \(\operatorname{Spec}(R[x_1^{\pm 1}, \dotsc , x_n^{\pm 1}])\).
Ask any toddler on the street.
For a group scheme \(G\) over \(S\), the character lattice of \(G\) is
An element \(X(G)\) is (unsurprisingly) called a character.
Characters of the torus over a field \(k\) are isomorphic to \({\mathbb Z}^n\). \(X({\mathbb G_m}^n) = {\mathbb Z}^n\).
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
where \(T\) is a split torus, \(\iota \) is a closed subgroup embedding and \(\phi \) is an fpqc homomorphism.
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.
Let \(T\) be a split torus. If \(H \subseteq T\) is an irreducible subgroup, then \(H\) is a split torus.
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.
For a finite dimensional representation of a torus \(T\) on \(W\), the character eigenspace of a character \(\chi \in X(T)\) is
The space decomposes into the direct sum of the character eigenspaces.
TODO
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.
Character lattice and one-parameter subgroup pairing.
\(N = \operatorname{Hom}(M, {\mathbb Z}) \cong {\mathbb Z}^n\). For \(u \in N\) we write \(\lambda ^u\) for the corresponding cocharacter.
1.1.2 The Definition of Affine Toric Variety
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
Given a finite set \({\mathcal A}= \{ a_1, \dotsc , a_s\} \subseteq M\), define \(\Phi _{\mathcal A}: T \to \mathbb {A}^s\) given by \(\Phi _{\mathcal A} (t) = (\chi ^{a_1} (t), \dotsc , \chi ^{a_s} (t))\).
\(Y_{\mathcal A}\) is the (Zariski) closure of \(\operatorname{im}\Phi _{\mathcal A}\) in \(\mathbb A^s\).
Proposition 1.1.8
TODO
1.1.4 Toric Ideals
The ideal of the affine toric variety \(Y_{\mathcal A}\) is
See [ 1 ] .
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.
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 \(x^\alpha - x^\beta \).
If \(S\) is an affine monoid, then \(\operatorname{Spec}(\Bbbk [S])\) is an affine toric variety.
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.
If \(S\) is an affine monoid, then the character lattice of \(\operatorname{Spec}(\Bbbk [S])\) is \({\mathbb Z}S\).
It is what it is.
If \(S\) is an affine monoid and \(\mathcal A\) is a finite set generating \(S\) as a monoid, then \(\operatorname{Spec}(\Bbbk [S]) = Y_{\mathcal A}\).
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 \(Y_{\mathcal A} = \mathbb V(\ker (\pi )) = \operatorname{Spec}(\Bbbk [x_1, \dotsc , x_s] / \ker (\pi )) = \operatorname{Spec}({\mathbb C}[S])\).
Torus action on semigroup algebra
1.1.5 Equivalence of Constructions
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 \(S\).
1.2 Cones and Affine Toric Varieties
1.2.1 Convex Polyhedral Cones
Fix a pair of dual real vector spaces \(M\) and \(N\).
For a set \(S \subseteq N\), the cone generated by \(S\), aka cone hull of \(S\), is
A polyhedral cone is a set that can be written as \(\operatorname{Cone}(S)\) for some finite set \(S\).
For a set \(S \subseteq N\), the convex hull of \(S\) is
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
Given a polyhedral cone \(\sigma \subseteq N\), its dual cone is defined by
.
If \(\sigma \) is polyhedral, then its dual \(\sigma ^\vee \) is polyhedral too.
Classic. See [ 3 ] maybe.
If \(\sigma _1, \sigma _2\) are two cones, then
Classic. See [ 3 ] maybe.
If \(\sigma \) is polyhedral, then \(\sigma ^{\vee \vee } = \sigma \).
Classic. See [ 3 ] maybe.
Given \(m \ne 0\) in \(M\), we get the hyperplane
and the closed half-space
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 \(\tau \prec \sigma \).
A dimension 1 face of a cone is called an edge.
A codimension 1 face of a cone is called a facet.
If \(\sigma \) is a polyhedral cone, then every face of \(\sigma \) is a polyhedral cone.
If \(\sigma \) is a polyhedral cone, then the intersection of two faces of \(\sigma \) is a face of \(\sigma \).
Classic. See [ 3 ] maybe.
A face of a face of a polyhedral cone \(\sigma \) is again a face of \(\sigma \).
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 \(v, w \in \tau \).
Classic. See [ 3 ] maybe.
If \(\sigma = H_{m_1}^+ \cap \dots \cap H_{m_s}^+\), then
Classic. See [ 3 ] maybe.
If \(\sigma \) is a full dimensional cone, then facets of \(\sigma \) are of the form \(H_m \cap \sigma \).
Classic. See [ 3 ] maybe.
Every proper face \(\tau \prec \sigma \) of a polyhedral cone \(\sigma \) is the intersection of the facets of \(\sigma \) containing \(\tau \).
Classic. See [ 3 ] maybe.
Given a cone \(\sigma \) and a face \(\tau \preceq \sigma \), the dual face to \(\tau \) is
If \(\tau \preceq \sigma \), then \(\tau ^* \preceq \sigma ^\vee \).
Classic. See [ 3 ] maybe.
If \(\tau \preceq \sigma \), then \(\tau ^{**} = \tau \).
Classic. See [ 3 ] maybe.
If \(\tau ' \preceq \tau \preceq \sigma \), then \(\tau ' \preceq \tau \).
Classic. See [ 3 ] maybe.
If \(\tau \preceq \sigma \), then
Classic. See [ 3 ] maybe.
1.2.3 Relative Interiors
The relative interior, aka intrinsic interior, of a cone \(\sigma \) is the interior of \(\sigma \) as a subset of its span.
For a cone \(\sigma \),
Classic. See [ 3 ] maybe.
If \(\tau \preceq \sigma \) and \(m \in \sigma ^\vee \), then
Classic. See [ 3 ] maybe.
If \(\sigma \) is a cone, then \(W := \sigma \cap (-\sigma )\) is a subspace. Furthermore, \(W = H_m \cap \sigma \) whenever \(m \in \operatorname{Relint}(\sigma ^\vee )\).
Classic. See [ 3 ] maybe.
1.2.4 Strong Convexity
A cone \(\sigma \) is salient, aka pointed or strongly convex, if \(\sigma \cap (-\sigma ) = \{ 0\} \).
The following are equivalent
\(\sigma \) is salient
\(\{ 0\} \preceq \sigma \)
\(\sigma \) contains no positive dimensional subspace
\(\dim \sigma ^\vee = \dim N\)
Classic. See [ 3 ] maybe.
1.2.5 Separation
Let \(\sigma _1, \sigma _2\) be polyhedral cones meeting along a common face \(\tau \). Then
for any \(m \in \operatorname{Relint}(\sigma _1^\vee \cap (-\sigma _2)^\vee )\).
See [ 1 ] .
1.2.6 Rational Polyhedral Cones
Let \(M\) and \(N\) be dual lattices with associated vector spaces \(M_{\mathbb R}:= M \otimes _{\mathbb Z}{\mathbb R}, N_{\mathbb R}:= N \otimes _{\mathbb Z}{\mathbb R}\).
A cone \(\sigma \subseteq N_{\mathbb R}\) is rational if \(\sigma = \operatorname{Cone}(S)\) for some finite set \(S \subseteq N\).
If \(\tau \preceq \sigma \) is a face of a rational cone, then \(\tau \) itself is rational.
Classic. See [ 3 ] maybe.
\(\sigma ^\vee \) is a rational cone iff \(\sigma \) is.
Classic. See [ 3 ] maybe.
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 \(\rho \).
The minimal generators of a rational cone \(\sigma \) are the ray generators of its edges.
A salient convex rational polyhedral cone is generated by its minimal generators.
Classic. See [ 3 ] maybe.
A salient rational polyhedral cone \(\sigma \) is regular, aka smooth, if its minimal generators form part of a \({\mathbb Z}\)-basis of \(N\).
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
If \(\sigma \subseteq N_{\mathbb R}\) is a polyhedral cone, then the lattice points
form a monoid.
\(S_\sigma \) is finitely generated as a monoid.
See [ 1 ] .
\(U_\sigma := \operatorname{Spec}{\mathbb C}[S_\sigma ]\) is an affine toric variety.
See [ 1 ] .
If \(\sigma \subseteq N_{\mathbb R}\) is salient of maximal dimension, then the irreducible elements of \(S_\sigma \) are precisely the minimal generators of \(\sigma ^\vee \).
See [ 1 ] .