4.4. Positivity properties of 𝕋 -Cartier divisors [02QE]

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4.4. Positivity properties of $\mathbb{T}$ -Cartier divisors

Let $\Sigma$ be a fan in $N_{\mathbb{R}}$ and $\Psi$ a virtual support function on $\Sigma$ . In this section, we will assume that $\Sigma$ is complete or, equivalently, that the variety $X_{\Sigma}$ is proper.

Many geometric properties of the pair $(X_{\Sigma},D_{\Psi})$ can be read directly from $\Psi$ . For instance, $\mathcal{O}(D_{\Psi})$ is generated by global sections if and only if the function $\Psi$ is concave, and the line bundle ${\mathcal{O}}(D_{\Psi})$ is ample if and only if $\Psi$ is strictly concave on $\Sigma$ . In the latter case, the fan $\Sigma$ agrees with the polyhedral complex $\Pi(\Psi)$ (Definition 3.34) and the pair $(X_{\Sigma},D_{\Psi})$ is completely determined by $\Psi$ . Thus, the variety $X_{\Sigma}$ is projective if and only if the fan $\Sigma$ is complete and regular (Definition 3.60).

We associate to $\Psi$ the subset of $M_{\mathbb{R}}$

\Delta_{\Psi}=\{x\in M_{\mathbb{R}}\mid\langle x,u\rangle\geq\Psi(u)\mbox{ for all }u\in N_{\mathbb{R}}\}.

This set is either empty or a lattice polytope. When ${\mathcal{O}}(D_{\Psi})$ is generated by global sections, the polytope $\Delta_{\Psi}$ agrees with $\operatorname{stab}(\Psi)$ , and $\Psi$ is the support function of $\Delta_{\Psi}$ .

The polytope $\Delta_{\Psi}$ encodes a lot of information about the pair $(X_{\Sigma},D_{\Psi})$ . For instance, we can read from it the space of global sections of $\mathcal{O}(D_{\Psi})$ . A monomial rational section $\chi^{m}\in\mathcal{K}_{X_{\Sigma}}$ , $m\in M$ , is a regular global section of $\mathcal{O}(D_{\Psi})$ if and only if $m\in\Delta_{\Psi}$ . Moreover, the set $\{\chi^{m}\}_{m\in M\cap\Delta_{\Psi}}$ is a $K$ -basis of the space of global sections $\Gamma(X_{\Sigma},\mathcal{O}(D_{\Psi}))$ . In the sequel we will see many more examples of this principle.

Proposition 4.37.

Let $D_{\Psi_{i}}$ , $i=1,\dots,n$ , be $\mathbb{T}$ -Cartier divisors on $X_{\Sigma}$ generated by their global sections. Then

(4.38)

(D_{\Psi_{1}}\cdot\dots\cdot D_{\Psi_{n}})=\operatorname{MV}_{M}(\Delta_{\Psi_{1}},\dots,\Delta_{\Psi_{n}}).

where $\operatorname{MV}_{M}$ denotes the mixed volume function associated to the Haar measure $\operatorname{vol}_{M}$ on $M_{\mathbb{R}}$ (Definition 3.109). In particular, for a $\mathbb{T}$ -Cartier divisor $D_{\Psi}$ generated by its global sections,

(4.39)

\deg_{D_{\Psi}}(X_{\Sigma})=(D_{\Psi}^{n})=n!\operatorname{vol}_{M}(\Delta_{\Psi}).

Proof.

This follows from [Oda88, Proposition 2.10]. ∎

Remark 4.40.

The intersection multiplicity and the degree in the above Proposition only depend on the isomorphism class of the line bundles ${\mathcal{O}}(D_{\Psi_{i}})$ and not on the $\mathbb{T}$ -Cartier divisors themselves. It is easy to check directly that the right-hand sides of (4.38) and (4.39) only depends on the isomorphism class of the line bundles. In fact, let $L$ be a toric line bundle generated by global sections and $s_{1}$ , $s_{2}$ two toric sections. For $i=1,2$ , set $D_{i}=\operatorname{div}(s_{i})$ and let $\Psi_{i}$ be the corresponding support function and $\Delta_{i}$ the associated polytope. Then $s_{2}=\chi^{m}s_{1}$ for some $m\in M$ . Thus $\Psi_{2}=\Psi_{1}-m$ and $\Delta_{2}=\Delta_{1}-m$ . Since the volume and the mixed volume are invariant under translation, we see that these formulae do not depend on the choice of sections.

Definition 4.41.

A polarized toric variety is a pair $(X_{\Sigma},D_{\Psi})$ , where $X_{\Sigma}$ is a toric variety and $D_{\Psi}$ is an ample $\mathbb{T}$ -Cartier divisor.

Polarized toric varieties can be classified in terms of their polytopes.

Theorem 4.42.

The correspondence $(X_{\Sigma},D_{\Psi})\mapsto\Delta_{\Psi}$ is a bijection between the set of polarized toric varieties and the set of lattice polytopes of dimension $n$ of $M$ . Two ample $\mathbb{T}$ -Cartier divisors $D_{\Psi}$ and $D_{\Psi^{\prime}}$ on a toric variety $X_{\Sigma}$ are rationally equivalent if and only if $\Delta_{\Psi^{\prime}}$ is the translated of $\Delta_{\Psi}$ by an element of $M$ .

Proof.

If $\Psi$ is a strictly concave function on $\Sigma$ , then $\Delta_{\Psi}$ is an $n$ -dimensional lattice polytope. Conversely, if $\Delta$ is a lattice polytope in $M_{\mathbb{R}}$ , then $\Psi_{\Delta}$ , the support function of $\Delta$ , is a strictly concave function on the complete fan $\Sigma_{\Delta}=\Pi(\Psi_{\Delta})$ (see examples 3.71 and 3.76). Therefore, the result follows from Theorem 4.18 and the construction of Remark 4.40. ∎

Remark 4.43.

When $D_{\Psi}$ is only generated by its global sections, the polytope $\Delta_{\Psi}$ may not determine the variety $X_{\Sigma}$ , but it does determine a polarized toric variety that is the image of $X_{\Sigma}$ by a toric morphism. Write $\Delta=\Delta_{\Psi}$ for short. Let $M(\Delta)$ be as in Notation 3.103 and choose $m\in\operatorname{aff}(\Delta)\cap M$ . Set $N(\Delta)=M(\Delta)^{\vee}$ . The translated polytope $\Delta-m$ has the same dimension as its ambient space $L_{\Delta}=M(\Delta)_{\mathbb{R}}$ . By the theorem above, it defines a complete fan $\Sigma_{\Delta}$ in $N(\Delta)_{\mathbb{R}}$ together with a support function $\Psi_{\Delta}\colon N(\Delta)\to\mathbb{R}$ . The projection $N\to N(\Delta)$ induces a toric morphism

\varphi\colon X_{\Sigma}\longrightarrow X_{\Sigma_{\Delta}},

the divisor $D_{\Psi_{\Delta}}$ is ample, and $D_{\Psi}=\varphi^{*}D_{\Psi_{\Delta}}+\operatorname{div}(\chi^{-m})$ .

Example 4.44.

The projective morphisms associated to $\mathbb{T}$ -Cartier divisors generated by global sections can also be made explicit in terms of the lattice points of the associated polytope. Consider a complete toric variety $X_{\Sigma}$ of dimension $n$ equipped with a $\mathbb{T}$ -Cartier divisor $D_{\Psi}$ generated by global sections. Let $m_{0},\dots,m_{r}\in\Delta_{\Psi}\cap M$ be such that $\operatorname{conv}(m_{0},\dots,m_{r})=\Delta_{\Psi}$ . These vectors determine an H-representation $\Psi=\min_{i=0,\dots,r}m_{i}$ . Let $H\colon N_{\mathbb{R}}\to\mathbb{R}^{r}$ be the linear map defined by $H(u)=(m_{i}(u)-m_{0}(u))_{i=1,\dots,r}$ . By Lemma 3.79, $\Psi=H^{\ast}\Psi_{\Delta^{r}}+m_{0}$ .

In $\mathbb{R}^{r}$ we consider the fan $\Sigma_{\Delta^{r}}$ , whose associated toric variety is $\mathbb{P}^{r}$ . One easily verifies that, for each $\sigma\in\Sigma$ , there is $\sigma^{\prime}\in\Sigma_{\Delta^{r}}$ with $H(\sigma)\subset\sigma^{\prime}$ . Let $p=(p_{0}:\dots:p_{r})$ be an arbitrary rational point of the principal open subset of $\mathbb{P}^{r}$ . The equivariant morphism $\varphi_{p,H}\colon X\to\mathbb{P}^{r}_{K}$ can be written explicitly as $(p_{0}\chi^{m_{0}}:\dots:p_{r}\chi^{m_{r}})$ . Moreover, $D_{\Psi}=\varphi_{p,H}^{\ast}D_{\Psi_{\Delta^{r}}}+\operatorname{div}(\chi^{-m_{0}})$ .

The orbits of a polarized toric variety $(X_{\Sigma},D_{\Psi})$ are in one-to-one correspondence with the faces of $\Delta_{\Psi}$ .

Proposition 4.45.

Let $\Sigma$ be a complete fan in $N_{\mathbb{R}}$ and $\Psi$ a strictly concave function on $\Sigma$ . The correspondence $F\mapsto O(\sigma_{F})$ is a bijection between the set of faces of $\Delta_{\Psi}$ and the set of the orbits under the action of $\mathbb{T}$ on $X_{\Sigma}$ .

Proof.

This follows from Example 3.71. ∎

Equation (4.25) gives a formula for the Weil divisor $[D_{\Psi}]$ in terms of the virtual support function $\Psi$ . When the line bundle $\mathcal{O}(D_{\Psi})$ is ample, we can interpret this formula in terms of the facets of the polytope $\Delta_{\Psi}$ .

Let $D_{\Psi}$ be an ample line bundle on $X_{\Sigma}$ . The polytope $\Delta_{\Psi}$ has maximal dimension $n$ . For each facet $F$ of $\Delta_{\Psi}$ , let $v_{F}$ be as in Notation 3.103. The ray $\tau_{F}=\mathbb{R}_{\geq 0}v_{F}$ is a cone of $\Sigma$ .

Proposition 4.46.

With the previous hypothesis,

\operatorname{div}(s_{\Psi})=[D_{\Psi}]=\sum_{F}-\langle v_{F},F\rangle V(\tau_{F}),

where the sum is over the facets $F$ of $\Delta$ .

Proof.

Since $\Psi$ is strictly concave on $\Sigma$ , the Legendre-Fenchel correspondence shows that the set of rays of the form $\tau_{F}$ agrees with the set $\Sigma^{1}$ . Moreover, $\Psi(v_{F})=\langle v_{F},F\rangle$ , because $\Psi$ is the support function of $\Delta$ . The proposition then follows from (4.25). ∎

For a $\mathbb{T}$ -Cartier divisor generated by global sections, we can interpret its intersection with the closure of an orbit, and its inverse image with respect to an equivariant morphism, in terms of direct and inverse images of concave functions.

Proposition 4.47.

Let $\Sigma$ be a complete fan in $N_{\mathbb{R}}$ and $\Psi\colon N_{\mathbb{R}}\to\mathbb{R}$ a support function on $\Sigma$ .

(1)

Let $\sigma\in\Sigma$ , $F_{\sigma}$ the associated face of $\Delta_{\Psi}$ , and $m_{\sigma}^{\prime}\in F_{\sigma}\cap M$ . Let $\pi_{\sigma}\colon N_{\mathbb{R}}\to N(\sigma)_{\mathbb{R}}$ be the natural projection. Then

(4.48) $(\Psi-m^{\prime}_{\sigma})(\sigma)=(\pi_{\sigma})_{\ast}(\Psi-m_{\sigma}^{\prime}).$

In particular, the restriction of $D_{\Psi-m^{\prime}_{\sigma}}$ to $V(\sigma)$ is given by the concave function $(\pi_{\sigma})_{\ast}(\Psi-m_{\sigma}^{\prime})$ . Moreover, the associated polytope is

(4.49) $\Delta_{(\Psi-m_{\sigma}^{\prime})(\sigma)}=F_{\sigma}-m_{\sigma}^{\prime}\subset M(\sigma)_{\mathbb{R}}=\sigma^{\bot}.$
(2)

Let $H\colon N^{\prime}\to N$ be a linear map and $H^{\vee}\colon M\to M^{\prime}$ its dual map, where $M^{\prime}=(N^{\prime})^{\vee}$ . Let $\Sigma^{\prime}$ be a fan in $N^{\prime}_{\mathbb{R}}$ such that, for each $\sigma^{\prime}\in\Sigma^{\prime}$ there is $\sigma\in\Sigma$ with $H(\sigma^{\prime})\subset\sigma$ , and let $p\in X_{\Sigma,0}(K)$ . Then

(4.50) $\varphi_{p,H}^{\ast}D_{\Psi}=D_{H^{\ast}\Psi},$

and the associated polytope is

(4.51) $\Delta_{H^{\ast}\Psi}=H^{\vee}(\Delta_{\Psi})\subset M^{\prime}_{\mathbb{R}}.$

Proof.

Equation (4.48) follows from (4.30), while equation (4.50) follows from Proposition 4.35. Then (4.49) and (4.51) follow from Proposition 3.78. ∎

As a consequence of the above construction, we can compute easily the degree of any orbit.

Corollary 4.52.

Let $\Sigma$ be a complete fan in $N_{\mathbb{R}}$ , $\Psi\colon N_{\mathbb{R}}\to\mathbb{R}$ a support function on $\Sigma$ , and $\sigma\in\Sigma$ a cone of dimension $n-k$ . Then

\deg_{D_{\Psi}}(V(\sigma))=k!\operatorname{vol}_{M(F_{\sigma})}(F_{\sigma}).

Proof.

In view of equations (4.49) and (4.39), it is enough to prove that $M(\sigma)=M(F_{\sigma})$ . But this follows from the fact that $L_{F_{\sigma}}=\sigma^{\perp}$ (see Notation 3.103). ∎

Example 4.53.

Let $\tau\in\Sigma^{n-1}$ . The degree of the curve $V(\tau)$ agrees with the lattice length of $F_{\tau}$ .

We will also need the toric version of the Nakai-Moishezon criterion.

Theorem 4.54.

Let $X_{\Sigma}$ be a proper toric variety and $D_{\Psi}$ a $\mathbb{T}$ -Cartier divisor on $X_{\Sigma}$ .

(1)
The following properties are equivalent:
1. (a)
  
  $D_{\Psi}$ is ample;
2. (b)
  
  $(D_{\Psi}\cdot C)>0$ for every curve $C$ in $X_{\Sigma}$ ;
3. (c)
  
  $(D_{\Psi}\cdot V(\tau))>0$ for every $\tau\in\Sigma^{n-1}$ .
(2)
The following properties are equivalent:
1. (a)
  
  $D_{\Psi}$ is generated by its global sections;
2. (b)
  
  $(D_{\Psi}\cdot C)\geq 0$ for every curve $C$ in $X_{\Sigma}$ ;
3. (c)
  
  $(D_{\Psi}\cdot V(\tau))\geq 0$ for every $\tau\in\Sigma^{n-1}$ .

Proof.

This follows from [Oda88, Theorem 2.18] for non singular toric varieties, and from [Mav00] for the general case. ∎

4.4. Positivity properties of 𝕋 -Cartier divisors [02QE]

4.4. Positivity properties of 𝕋\mathbb{T}-Cartier divisors

Proposition 4.37.

Proof.

Remark 4.40.

Definition 4.41.

Theorem 4.42.

Proof.

Remark 4.43.

Example 4.44.

Proposition 4.45.

Proof.

Proposition 4.46.

Proof.

Proposition 4.47.

Proof.

Corollary 4.52.

Proof.

Example 4.53.

Theorem 4.54.

Proof.

4.4. Positivity properties of $\mathbb{T}$ -Cartier divisors