Let us work over an arbitrary field $k$.
Toric varieties
Definition 1. A toric variety is a split torus $T/k$ together with an open embedding $T \hookrightarrow X$ into an irreducible variety $X/k$ such that the translation action of $T$ on itself extends to an action on $X$. For toric varieties $(T, X)$ and $(S, Y)$, a toric morphism is a morphism of tori $T \to S$ together with a morphism $X \to Y$ that is equivariant for $T \to S$.
Example 2. For $T = \mathbb{G}_ m^2$, we can add in the two axes and consider $X = \mathbb{A}^2$. We can also compactify it to $X = \mathbb{P}^2$ or $X = \mathbb{P}^1 \times \mathbb{P}^1$.
Our goal today is to classify normal toric varieties. We start with a useful theorem that allows us to reduce to understanding affines.
Theorem 3 (Sumihiro, 1974). Let $X$ be a normal variety with an action of a split torus $T$. Then $X$ can be covered by $T$-stable affine opens.
Remark 4. Normality is essential. A counterexample is $\mathbb{P}^1$ with the usual $\mathbb{G}_ m$-action, but with $0$ and $\infty$ glued together. Then the only $T$-stable neighborhood of this point is the entire curve, which is not affine.
The upshot is that we can glue these $T$-stable affine charts together the understand the toric variety. We have $T = \Spec k[X^\ast(T)]$, and let’s say that $T \subseteq U = \Spec A$. Because $X$ is a variety, $T$ is dense inside it, and this means that $A \subseteq k[X^\ast(T)]$. Next, because $T$ acts on $A$, we can split it off into representations and write $$ A = \bigoplus_{\lambda \in \operatorname{wt}(A) \subseteq X^\ast(T)} k \subseteq \bigoplus_{\lambda \in X^\ast(T)} = k[X^\ast(T)]. $$ Because $A$ is a $k$-algebra, we see that $\operatorname{wt}(A) \subseteq X^\ast(T)$ is an additive submonoid, and because $A$ is normal, it is saturated in the sense that $x^n \in \operatorname{wt}(A)$ implies $x \in \operatorname{wt}(A)$. It also has to be finitely generated because it is a variety, and also has to generate $X^\ast(T)$ as an abelian group.
Definition 5. Let $V$ be a finite-dimensional $\mathbb{R}$-vector space. A convex cone is a closed subset $C \subseteq V$ containing the origin, closed under nonnegative linear combinations. We say that $C$ is non-degenerate if it is not contained in a proper $\mathbb{R}$-vector subspace, and strongly convex if it does not contain a line.
For $C \subseteq V$ a convex cone, we can define its dual $$ C^\vee = \lbrace x \in V^\vee : \langle x, y \rangle \ge 0 \text{ for all } y \in C \rbrace \subseteq V^\vee. $$ This defines a inclusion-reversing bijection between convex cones of $V$ and convex cones of $V^\vee$. We see that strongly convex corresponds to non-degenerate under this duality.
Definition 6. A rational polyhedral cone is a closed subset $C \subseteq X^\ast(T)_ \mathbb{R}$ of the form $$ C = \lbrace x \in X^\ast(T)_ \mathbb{R} : \langle \alpha_1, x \rangle, \dotsc, \langle \alpha_n, x \rangle \ge 0 \rbrace $$ for $\alpha_1, \dotsc, \alpha_n \in X^\ast(T)$.
We easily see that rational polyhedral cones for $X^\ast(T)$ are dual to rational polyhedral cones for $X_\ast(T)$. We also observe that for any rational polyhedral cone $C \subseteq X^\ast(T)_ \mathbb{R}$, the additive monoid $C \cap X^\ast(T)$ is finitely generated.
Definition 7. For a strongly convex rational polyhedral cone $\sigma \subseteq X_\ast(T)_ \mathbb{R}$, we define the corresponding affine toric variety $$ U_\sigma = \Spec k[\sigma^\vee \cap X^\ast(T)]. $$
Given any inclusion $\sigma \subseteq \tau$, we have $\sigma^\vee \supseteq \tau^\vee$ and so we get a map $U_\sigma \to U_\tau$.
Lemma 8. This map $U_\sigma \to U_\tau$ is an open embedding if and only if $\sigma$ is a face of $\tau$.
Classification of normal toric varieties
We can now start gluing these affine toric varieties.
Lemma 9. If $U_\sigma, U_\tau \subseteq X$ are $T$-equivariant affine opens, then $U_\sigma \cap U_\tau$ is the affine open subset $U_{\sigma \cap \tau}$. In particular, $\sigma \cap \tau$ is a face of both $\sigma$ and $\tau$.
More generally, we have the following.
Lemma 10. Let $(T, X)$ be a toric variety, and let $U_\sigma, U_\tau \to X$ be toric morphisms. Then $U_\sigma \times_X U_\tau \cong U_{\sigma \cap \tau}$ as $T$-toric varieties.
To see this, we note that $U_\sigma \cap U_\tau = U_\kappa$ for some strongly convex rational polyhedral cone $\kappa \subseteq \sigma \cap \tau$. On the other hand, $U_\kappa \to U_\sigma \times U_\tau$ has to be a closed embedding. This shows that $\kappa^\vee = \sigma^\vee + \tau^\vee$, and so dualizing this gives $\kappa = \sigma \cap \tau$.
Definition 11. A fan is a finite collection $\Delta$ of strongly convex rational polyhedral cones in $X_\ast(T)_ \mathbb{R}$ such that
- for $\sigma \in \Delta$, every face of $\sigma$ is also in $\Delta$,
- for $\sigma, \tau \in \Delta$, the intersection $\sigma \cap \tau$ is a face of both $\sigma$ and $\tau$.
Theorem 12. There is a correspondence between fans $\Delta$ for $X_\ast(T)$ and normal toric varieties $X(\Delta)$ for $T$.
Given a normal toric variety, we can look at the set of all $T$-stable affine opens and then they form a fan. Conversely, if we have a fan, we can glue the corresponding affine toric varieties and glue them along face maps, and using the formula $U_\sigma \cap U_\tau = U_{\sigma \cap \tau}$ we can check that this both well-defined and is separated. Finally, we check that the affine toric charts in $X(\Delta)$ have to be of the form $U_\sigma \subseteq X(\Delta)$, and this implies that $\sigma$ maps into some $\tau \in \Delta$, which means that $\sigma$ is a face of $\Delta$ hence $\sigma \in \Delta$ as well.
Proposition 13. Let $T \to S$ be a map of split tori, and let $\Delta_T$ and $\Delta_S$ be fans for $T$ and $S$. Then there exists a toric morphism $X(\Delta_T) \to X(\Delta_S)$ if and only if $X_\ast(T) \to X_\ast(S)$ maps every cone in $\Delta_T$ into a cone in $\Delta_S$.
Example 14. Discuss the examples of
- the fan for $\mathbb{G}_ m^2$,
- the fan for $\mathbb{A}^2$,
- the fan for $\mathbb{P}^2$,
- the fan for $\mathbb{P}^1 \times \mathbb{P}^1$,
- Hirzebruch surfaces and blowups at $T$-fixed points?
Properties
Here are general properties.
- Normal toric varieties are Cohen–Macaulay.
- A toric variety $X(\Delta)$ is smooth if and only if it is regular if and only if for every $\sigma \in \Delta$ satisfies the property that $\sigma \cap X_\ast(T) \cong \mathbb{Z}_ {\ge 0}^k$ as additive monoids.
- The toric variety $X(\Delta)$ is proper if and only if $\operatorname{supp}(\Delta) = X_\ast(T)_ \mathbb{R}$. In this case, we say that $\Delta$ is complete.
- More generally, a toric map $X(\Delta) \to X(\Delta^\prime)$ is proper if and only if $\operatorname{supp}(\Delta)$ is the preimage of $\operatorname{supp}(\Delta^\prime)$.
- The $T$-orbits of $X(\Delta)$ corresponds to elements of $\Delta$.
- Consider the abelian group $\operatorname{Div}_ T(X(\Delta))$ of functions $f \colon \operatorname{supp}(\Delta) \to \mathbb{R}$ with the property that for every $\sigma \in \Delta$ there exists an element $\alpha \in X^\ast(T)$ such that $f \vert_\sigma = \alpha \vert_\sigma$. Then we have $$ X^\ast(T) \to \operatorname{Div}_ T(X(\Delta)) \to \operatorname{Pic}(X(\Delta)) \to 0. $$
- Assuming that $\operatorname{supp}(\Delta) = X_\ast(T)_ \mathbb{R}$, this corresponds to a polytope $P_D$ up to translation. The line bundle $\mathscr{O}(D)$ is generated by global sections if and only if $P_D$ is convex. It is ample if and only if $P_D$ is strictly convex.
- In the convex case, the dimension of $H^0(X, \mathscr{O}(D))$ is the number of points in $P(D) \cap X^\ast(T)$.