5 Toric varieties
5.1 Toric varieties
In this section, we define toric varieties and toric morphisms.
Let \(k\) be a field. Let \(T\) be a torus over \(k\). A toric variety structure on a scheme \(X\) over \(k\) consists the following data:
a torus \(T\) over \(k\),
a group action \(T \times X \to X\) over \(k\).
a dominant open immersion \(i : T \hookrightarrow X\) over \(k\) that is \(T\)-equivariant.
Let \(k\) be a field. Let \(T_1, T_2\) be tori over \(k\). Let \(X_1, X_2\) be toric varieties with torus \(T_1, T_2\) respectively. A toric morphism from \(X_1\) to \(X_2\) is the data of a \(k\)-morphism \(X_1 \to X_2\) and a \(k\)-group homomorphism \(T_1 \to T_2\) that commute with the embeddings \(T_1 \to X_1, T_2 \to X_2\) and the actions. A toric isomorphism from \(X_1\) to \(X_2\) is the data of two isomorphisms \(X_1 \cong X_2\) and \(T_1 \cong T_2\) that commute with the embeddings \(T_1 \to X_1, T_2 \to X_2\).
5.2 Affine toric varieties and affine monoids
In this section, we construct affine toric varieties from affine monoids, and show all affine toric varieties arise from affine monoids in this way.
5.2.1 Toric varieties from affine monoids
Let \(k\) be a field. Let \(G\) be a finitely generated free abelian group. Let \(M\) be an affine monoid with Grothendieck group \(G\). Then \(D_k(M)\) is an affine toric variety over \(k\) with torus \(D_k(G)\).
The map \(D_k(G) \hookrightarrow D_k(M)\) is given by the embedding \(M \hookrightarrow G\).
We identify \(D_k(G) \cong \operatorname{Spec}k[G], D_k(M) \cong \operatorname{Spec}k[M]\) using Proposition 4.2.3.
By Proposition 2.3.12, \(k[G]\) is a localization of \(k[M]\). Therefore the map \(\operatorname{Spec}k[G] \hookrightarrow \operatorname{Spec}k[M]\) is an open immersion.
This open immersion is dominant since \(\operatorname{Spec}k[M]\) is irreducible as \(k[M]\) is a domain (Proposition 2.2.4).
The group action \(D_k(G) \times D_k(M) \to D_k(M)\) comes from pulling back along \(D_k(G) \hookrightarrow D_k(M)\) the left action \(D_k(M) \times D_k(M) \to D_k(M)\) using Proposition 1.2.5.
5.2.2 Essential surjectivity from affine monoids to affine toric varieties
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
There is a torus action on the semigroup algebra \({\mathbb C}[M]\): given \(t\in T_N\) and \(f\in {\mathbb C}[M]\) define
Let \(A \subseteq {\mathbb C}[M]\) be a stable subspace, then
TODO
Let \(k\) be a field. Let \(T\) be a torus over \(k\). Let \(V\) be a toric variety with torus \(k\). The characters \(X(V)\) of \(V\) are defined as the intersection of \(X(T)\) with the image of the map \(k[V] \to k[T]\) of coordinate rings induced by the embedding \(T \hookrightarrow V\).
Let \(k\) be a field. Let \(T\) be a torus over \(k\). Let \(V\) be a toric variety with torus \(k\). Then \(X(V)\) is an affine monoid.
TODO
Let \(k\) be a field. Let \(T\) be a torus over \(k\). Let \(V\) be a toric variety with torus \(k\). Then there exists a torus isomorphism \(V \cong D_k(X(V))\).
TODO
5.3 Affine toric varieties and toric ideals
In this section, we define toric ideals, show that one can construct toric varieties from them and that all toric varieties arise in this way.
5.3.1 Toric ideals and affine monoids
Let \(R\) be a ring. Let \(G\) be a free abelian group and \(M\) an affine monoid whose Grothendieck group is \(G\). Let \(L \le G\) be a sublattice. The lattice ideal of \(L\) is the \(R\)-ideal of \(R[M]\) defined by
Let \(R\) be a ring. Let \(M\) be an affine monoid. A toric ideal is a prime lattice \(R\)-ideal of \(R[M]\).
An ideal is toric if and only if it’s prime and generated by binomials \(X^\alpha - X^\beta \).
A toric ideal is prime and generated by binomials by definition.
Assume \(I\) is prime and generated by \(X^\alpha - X^\beta \) ranging over \((\alpha , \beta ) \in S\) for some set \(S \subseteq M \times M\).
Note first that \(I\) doesn’t contain any monomial. Indeed, \(I\) is contained in the kernel of the map \(R[M] \to R\) given by \(X^m \mapsto 1\).
Since \(I\) is prime, this means that
In particular, if \(\alpha _1 - \beta _1 = \alpha _2 - \beta _2\), then Since \(I\) is prime, this means that
Now, we claim that \(I = I_L\) where \(L \le G\) is given by
Clearly, \(I \subseteq I_L\).
For the other direction, assume \(\delta , \varepsilon \in M, \delta - \varepsilon \in L\). Let’s prove \(X^\delta - X^\varepsilon \in I\) by induction on \(\delta - \varepsilon \in L\):
If \(\delta - \varepsilon = 0\), then \(X^\delta - X^\varepsilon = 0 \in I\).
If \(\delta _1 - \varepsilon _1 = \varepsilon _2 - \delta _2\) and \(X^{\delta _2} - X^{\varepsilon _2} \in I\), then
\[ X^{\delta _1} - X^{\varepsilon _1} \in I \iff X^{\varepsilon _2} - X^{\delta _2} \iff X^{\delta _1} - X^{\varepsilon _1} \]by the remark, and this holds by assumption.
If \(\delta - \varepsilon = \alpha - \beta \) where \((\alpha , \beta ) \in S\), then
\[ X^\delta - X^\varepsilon \in I \iff X^\alpha - X^\beta \in I \]by the remark, and this holds by assumption.
Assume \(\delta _1, \delta _2, \varepsilon _1, \varepsilon _2\) are such that \(X^{\delta _1} - X^{\varepsilon _1}, X^{\delta _2} - X^{\varepsilon _2} \in I\). Then
\[ X^{\delta _1 + \delta _2} - X^{\varepsilon _1 + \varepsilon _2} = (X^{\delta _1} - X^{\varepsilon _1})X^{\delta _2} + X^{\varepsilon _1}(X^{\delta _2} - X^{\varepsilon _2}) \in I. \]
Let \(k\) be a field. Let \(V\) be a toric variety over \(k\). Let \(i : V \hookrightarrow \mathbb {A}^n\) be a closed toric embedding. Then the vanishing ideal of \(i\) is toric.
TODO