4.7. Positivity on toric schemes [02S8]

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4.7. Positivity on toric schemes

The relationship between the positivity of the line bundle and the concavity of the virtual support function can be extended to the case of toric schemes over a DVR. In particular, we have the following version of the Nakai-Moishezon criterion.

Theorem 4.95.

Let $\Pi$ be a complete SCR complex in $N_{\mathbb{R}}$ and ${\mathcal{X}}_{\Pi}$ its associate toric scheme over $S$ . Let $\psi$ be an H-lattice function on $\Pi$ and $D_{\psi}$ the corresponding $\mathbb{T}$ -Cartier divisor on ${\mathcal{X}}_{\Pi}$ .

(1)
The following properties are equivalent:
1. (a)
  
  $D_{\psi}$ is ample;
2. (b)
  
  $D_{\psi}\cdot C>0$ for every vertical curve $C$ contained in $X_{\Pi,o}$ ;
3. (c)
  
  $D_{\psi}\cdot V(\Lambda)>0$ for every $(n-1)$ -dimensional polyhedron $\Lambda\in\Pi$ ;
4. (d)
  
  The function $\psi$ is strictly concave on $\Pi$ .
(2)
The following properties are equivalent:
1. (a)
  
  $D_{\psi}$ is generated by global sections;
2. (b)
  
  $D_{\psi}\cdot C\geq 0$ for every vertical curve $C$ contained in $X_{\Sigma,o}$ ;
3. (c)
  
  $D_{\psi}\cdot V(\Lambda)\geq 0$ for every $(n-1)$ -dimensional polyhedron $\Lambda\in\Pi$ ;
4. (d)
  
  The function $\psi$ is concave.

Proof.

In both cases, the fact that (a) implies (b) and that (b) implies (c) is clear. The fact that (c) implies (d) follows from equation (4.93). The fact that (1d) implies (1a) is [KKMS73, §IV.3(k)].

Finally, we prove that (2d) implies (2a). Let $\psi$ be an H-lattice concave function. Each pair $(m,l)\in{\widetilde{M}}$ defines a rational section $\varpi^{l}\chi^{m}s_{\psi}$ of $D_{\psi}$ . The section is regular if and only if the function $m(u)+l$ lies above $\psi$ . Moreover, for a polyhedron $\Lambda\in\Pi$ , this section does not vanish on ${\mathcal{X}}_{\Lambda}$ if and only if $\psi(u)=m(u)+l$ for all $u\in\Lambda$ . Therefore, the affine pieces of the graph of $\psi$ define a set of global sections that generate $\mathcal{O}(D_{\psi})$ . ∎

Definition 4.96.

We will say that a $\mathbb{T}$ -Cartier divisor on a toric scheme is semipositive if it is generated by global sections. Let $X_{\Sigma}$ be a proper toric variety over $K$ and let $D_{\Psi}$ be a $\mathbb{T}$ -Cartier divisor generated by global sections. A toric model $({\mathcal{X}},D,e)$ is called semipositive if $D$ is semipositive.

Observe that, by Theorem 4.95, a toric model is semipositive if the associated metric is semipositive as in Definition 2.26. Equivalence classes of semipositive toric models are classified by rational concave functions.

Theorem 4.97.

Let $\Sigma$ be a complete fan in $N_{\mathbb{R}}$ . Let $\Psi$ be a support function on $\Sigma$ . Then the correspondence of Theorem 4.81 induces a bijective correspondence between the set of rational piecewise affine concave functions $\psi$ with $\operatorname{rec}(\psi)=\Psi$ and the set of equivalence classes of semipositive toric models of $(X_{\Sigma},D_{\Psi})$ over $S$ .

Proof.

Let $({\mathcal{X}},D,e)$ be a semipositive toric model. By Theorem 4.81, to the pair $({\mathcal{X}},D)$ corresponds a pair $(\Pi,\psi^{\prime})$ , where $\psi^{\prime}$ is an H-lattice function on $\Pi$ , $\operatorname{rec}(\Pi)=\Sigma$ and $\operatorname{rec}(\psi^{\prime})=e\Psi$ . By Theorem 4.95, the function $\psi^{\prime}$ is concave. We put $\psi=\frac{1}{e}\psi^{\prime}$ . It is clear that equivalent models produce the same function.

Conversely, let $\psi$ be a rational piecewise affine concave function. Let $\Pi^{\prime}=\Pi(\psi)$ . This is a rational polyhedral complex. Let $\Sigma^{\prime}=\operatorname{rec}(\Pi^{\prime})$ . This is a conic rational polyhedral complex. By Proposition 3.72, $\Sigma^{\prime}=\Pi(\Psi)$ . Since $\Psi$ is a support function on $\Sigma$ , we deduce that $\Sigma$ is a refinement of $\Sigma^{\prime}$ . Put $\Pi=\Pi^{\prime}\cdot\Sigma$ (Definition 3.10). Since $\Pi^{\prime}$ is a rational polyhedral complex and $\Sigma$ is a fan, then $\Pi$ is an SCR polyhedral complex. Moreover, by Lemma 3.11, we have

\operatorname{rec}(\Pi)=\operatorname{rec}(\Pi^{\prime}\cdot\Sigma)=\operatorname{rec}(\Pi^{\prime})\cdot\operatorname{rec}(\Sigma)=\Sigma^{\prime}\cdot\Sigma=\Sigma.

Let $e>0$ be an integer such that $e\psi$ is an H-lattice function. Then $({\mathcal{X}}_{\Pi},D_{e\psi},e)$ is a toric model of $(X_{\Sigma},D_{\Psi})$ . Both procedures are inverse of each other. ∎

Recall that, for toric varieties over a field, a $\mathbb{T}$ -Cartier divisor generated by global sections can be determined, either by the support function $\Psi$ or by its stability set $\Delta_{\Psi}$ . In the case of toric schemes over a DVR, if $\psi$ is a concave rational piecewise affine function on $\Pi$ and $\Psi=\operatorname{rec}(\psi)$ , then the stability set of $\psi$ agrees with the stability set of $\Psi$ . Then the equivalence class of toric models determined by $\psi$ is also determined by the Legendre-Fenchel dual function $\psi^{\vee}$ .

Corollary 4.98.

Let $\Sigma$ be a complete fan in $N_{\mathbb{R}}$ and $\Psi$ a support function on $\Sigma$ . There is a bijection between equivalence classes of semipositive toric models of $(X_{\Sigma},D_{\Psi})$ and rational piecewise affine concave functions on $M_{\mathbb{R}}$ , with effective support $\Delta_{\Psi}$ .

Proof.

This follows from Theorem 4.97, Proposition 3.75 and Proposition 3.77. ∎

When $D_{\psi}$ is generated by global sections, that is, when $\psi$ is concave, we can interpret its restriction to toric orbits in terms of direct and inverse images of concave functions.

Proposition 4.99.

Let $\Pi$ be a complete SCR polyhedral complex in $N_{\mathbb{R}}$ and $\psi$ an H-lattice concave function on $\Pi$ . Set $\Sigma=\operatorname{rec}(\Pi)$ and $\Psi=\operatorname{rec}(\psi)$ . Let $\sigma\in\Sigma$ and $m_{\sigma}\in M$ such that $\Psi|_{\sigma}=m_{\sigma}|_{\sigma}$ . Let $\pi_{\sigma}\colon N_{\mathbb{R}}\to N(\sigma)_{\mathbb{R}}$ be the projection and $\pi^{\vee}_{\sigma}\colon M(\sigma)_{\mathbb{R}}\to M_{\mathbb{R}}$ the dual inclusion. Then

(4.100)

(\psi-m_{\sigma})(\sigma)=(\pi_{\sigma})_{\ast}(\psi-m_{\sigma}),

Hence the restriction of the divisor $D_{\psi-m_{\sigma}}$ to ${\mathcal{V}}(\sigma)$ corresponds to the H-lattice concave function $(\pi_{\sigma})_{\ast}(\psi-m_{\sigma})$ . Dually,

(4.101)

(\psi-m_{\sigma})(\sigma)^{\vee}=(\pi^{\vee}_{\sigma}+m_{\sigma})^{\ast}\psi^{\vee}.

In other words, the Legendre-Fenchel dual of $(\psi-m_{\sigma})(\sigma)$ is the restriction of $\psi^{\vee}$ to the face $F_{\sigma}$ translated by $-m_{\sigma}$ .

Proof.

For equation (4.100), we suppose without loss of generality that $m_{\sigma}=0$ , and hence $\Psi|_{\sigma}=0$ . Let $u\in N(\sigma)_{\mathbb{R}}$ . Then, the function $\psi|_{\pi^{-1}_{\sigma}(u)}$ is concave. Let $\Lambda\in\Pi$ such that $\operatorname{rec}(\Lambda)=\sigma$ and $\pi^{-1}_{\sigma}(u)\cap\Lambda\not=\emptyset$ . Then, $\pi^{-1}_{\sigma}(u)\cap\Lambda$ is a polyhedron of maximal dimension in $\pi^{-1}_{\sigma}(u)$ . The restriction of $\psi$ to this polyhedron is constant and, by (4.88), agrees with $\psi(\sigma)(u)$ . Therefore, by concavity,

(\pi_{\sigma})_{\ast}\psi(u)=\max_{v\in\pi^{-1}_{\sigma}(u)}\psi(v),

agrees with $\psi(\sigma)(u)$ . Thus we obtain equation (4.100). Equation (4.101) follows from the previous equation and Proposition 3.78(2). To prove equation (4.101) when $m_{\sigma}\not=0$ we use Proposition 3.40(4). ∎

We now consider the case of a vertical orbit. For a function $\psi$ as before, with $\Psi=\operatorname{rec}(\psi)$ , we denote by $\operatorname{c}(\psi)\colon{\widetilde{N}}_{\mathbb{R}}\to\underline{\mathbb{R}}$ the concave function given by

\operatorname{c}(\psi)(u,r)=\begin{cases}r\psi(u/r),&\text{ if }r>0,\\ \Psi(u),&\text{ if }r=0,\\ -\infty,&\text{ if }r<0.\end{cases}

The function $\operatorname{c}(\psi)$ is a support function on $\operatorname{c}(\Pi)$ .

Lemma 4.102.

The stability set of $\operatorname{c}(\psi)$ is the epigraph $\operatorname{epi}(-\psi^{\vee})\subset{\widetilde{M}}_{\mathbb{R}}$ .

Proof.

The H-representation of $\operatorname{c}(\psi)$ is

	$\displaystyle{\operatorname{dom}}(\operatorname{c}(\psi))$	$\displaystyle=\{(u,r)\in{\widetilde{N}}_{\mathbb{R}}\mid r\geq 0\},$
	$\displaystyle\operatorname{c}(\psi)(u,r)$	$\displaystyle=\min_{\Lambda}(m_{\Lambda}(u)+l_{\Lambda}r).$

By Proposition 3.64

\operatorname{stab}(\operatorname{c}(\psi))=\mathbb{R}_{\geq 0}(0,1)+\operatorname{conv}(\{(m_{\Lambda},l_{\Lambda})\}_{\Lambda\in\Pi}).

Furthermore, by the same proposition, for $x\in\operatorname{stab}(\psi)$ ,

\psi^{\vee}(x)=\sup\left\{\sum_{\Lambda}-\lambda_{\Lambda}l_{\Lambda}\bigg|\lambda_{\Lambda}\geq 0,\sum_{\Lambda}\lambda_{\Lambda}=1,\sum_{\Lambda}\lambda_{\Lambda}m_{\Lambda}=x\right\}.

Hence $\operatorname{epi}(-\psi^{\vee})=\mathbb{R}_{\geq 0}(0,1)+\operatorname{conv}(\{(m_{\Lambda},l_{\Lambda})\}_{\Lambda\in\Pi}),$ which proves the statement. ∎

Proposition 4.103.

Let $\Pi$ and $\psi$ be as before and let $\Lambda\in\Pi$ . Let $m_{\Lambda}\in M$ and $l_{\Lambda}\in\mathbb{Z}$ be such that $\psi|_{\Lambda}=(m_{\Lambda}+l_{\Lambda})|_{\Lambda}$ . Let ${\widetilde{\pi}}_{\Lambda}\colon{\widetilde{N}}_{\mathbb{R}}\to{\widetilde{N}}(\Lambda)_{\mathbb{R}}$ be the projection, and ${\widetilde{\pi}}^{\vee}_{\Lambda}\colon{\widetilde{M}}(\Lambda)_{\mathbb{R}}\to{\widetilde{M}}_{\mathbb{R}}$ the dual map. Then

(4.104)

(\psi-m_{\Lambda}-l_{\Lambda})(\Lambda)=({\widetilde{\pi}}_{\Lambda})_{\ast}(\operatorname{c}(\psi-m_{\Lambda}-l_{\Lambda})).

Moreover, this is a support function on the fan $\Pi(\Lambda)$ . Its stability set is the polytope $\Delta_{\psi,\Lambda}:=({\widetilde{\pi}}^{\vee}_{\Lambda}+(m_{\Lambda},l_{\Lambda}))^{-1}\operatorname{epi}(-\psi^{\vee})$ . Hence, the restriction of the divisor $D_{\psi-m_{\Lambda}-l_{\Lambda}}$ to the variety $V(\Lambda)$ is the divisor associated to the support function of $\Delta_{\psi,\Lambda}$

Proof.

To prove equation (4.104) we may assume that $m_{\Lambda}=0$ and $l_{\Lambda}=0$ . Let $u\in{\widetilde{N}}(\Lambda)_{\mathbb{R}}$ . Then, the function $\operatorname{c}(\psi)|_{{\widetilde{\pi}}^{-1}_{\Lambda}(u)}$ is concave. Let $\Lambda^{\prime}\in\Pi$ such that $\Lambda$ is a face of $\Lambda^{\prime}$ and ${\widetilde{\pi}}^{-1}_{\Lambda}(u)\cap\operatorname{c}(\Lambda^{\prime})\not=\emptyset$ . Then, ${\widetilde{\pi}}^{-1}_{\Lambda}(u)\cap\operatorname{c}(\Lambda^{\prime})$ is a polyhedron of maximal dimension of ${\widetilde{\pi}}^{-1}_{\Lambda}(u)$ and the restriction of $\operatorname{c}(\psi)$ to this polyhedron is constant and, by equation (4.90), agrees with $\psi(\Lambda)(u)$ . Therefore, by concavity,

({\widetilde{\pi}}_{\Lambda})_{\ast}\operatorname{c}(\psi)(u)=\max_{v\in\pi^{-1}_{\sigma}(u)}\operatorname{c}(\psi)(v),

agrees with $\psi(\Lambda)(u)$ . This proves equation (4.104).

Back in the general case when $m_{\Lambda}$ and $l_{\Lambda}$ may be different from zero, by Proposition 3.78, Proposition 3.40(4) and Lemma 4.102 we have

	$\displaystyle\operatorname{stab}(({\widetilde{\pi}}_{\Lambda})_{\ast}(\operatorname{c}(\psi-m_{\Lambda}-l_{\Lambda})))$	$\displaystyle=({\widetilde{\pi}}^{\vee}_{\Lambda})^{-1}\operatorname{stab}(\operatorname{c}(\psi-m_{\Lambda}-l_{\Lambda}))$
		$\displaystyle=({\widetilde{\pi}}^{\vee}_{\Lambda})^{-1}(\operatorname{stab}(\operatorname{c}(\psi))-(m_{\Lambda},l_{\Lambda}))$
		$\displaystyle=({\widetilde{\pi}}^{\vee}_{\Lambda}+(m_{\Lambda},l_{\Lambda}))^{-1}\operatorname{stab}(\operatorname{c}(\psi))$
		$\displaystyle=({\widetilde{\pi}}^{\vee}_{\Lambda}+(m_{\Lambda},l_{\Lambda}))^{-1}\operatorname{epi}(-\psi^{\vee}).$

The remaining statements are clear. ∎

We next interpret the above result in terms of dual polyhedral complexes. Let $\Pi(\psi)$ and $\Pi(\psi^{\vee})$ be the pair of dual polyhedral complexes associated to $\psi$ . Since $\psi$ is piecewise affine on $\Pi$ , then $\Pi$ is a refinement of $\Pi(\psi)$ . For each $\Lambda\in\Pi$ we will denote by $\overline{\Lambda}\in\Pi(\psi)$ the smallest element of $\Pi(\psi)$ that contains $\Lambda$ . It is characterized by the fact that $\operatorname{ri}(\Lambda)\cap\operatorname{ri}(\overline{\Lambda})\not=\emptyset.$ Let $\Lambda^{\ast}\in\Pi(\psi^{\vee})$ be the polyhedron $\Lambda^{\ast}={\mathcal{L}}\psi(\overline{\Lambda})$ . This polyhedron agrees with $\partial\psi(u_{0})$ for any $u_{0}\in\operatorname{ri}(\Lambda)$ . Then the function $\psi^{\vee}|_{\Lambda^{\ast}}$ is affine. The polyhedron $\Lambda^{\ast}-m_{\Lambda}$ is contained in $M(\Lambda)_{\mathbb{R}}$ . The polyhedron

{\widetilde{\Lambda^{\ast}}}=\{(x,-\psi^{\vee}(x))|x\in\Lambda^{\ast}\}

is a face of $\operatorname{epi}(-\psi^{\vee})$ and it agrees with the intersection of the image of $\pi^{\vee}_{\Lambda}+(m_{\Lambda},l_{\Lambda})$ with this epigraph. We consider the commutative diagram of lattices

where $\pi^{\vee}_{\Lambda}$ is the inclusion $M(\Lambda)\subset M$ , and the corresponding commutative diagram of real vector spaces obtained by tensoring with $\mathbb{R}$ . This diagram induces a commutative diagram of polytopes

where all the arrows are isomorphisms.

In other words, the polytope $\Delta_{\psi,\Lambda}$ associated to the restriction of $D_{\psi-m_{\Lambda}-l_{\Lambda}}$ to $V(\Lambda)$ is obtained as follows. We include ${\widetilde{M}}(\Lambda)_{\mathbb{R}}$ in ${\widetilde{M}}_{\mathbb{R}}$ throughout the affine map ${\widetilde{\pi}}^{\vee}_{\Lambda}+(m_{\Lambda},l_{\lambda})$ . The image of this map intersects the polyhedron $\operatorname{epi}(-\psi^{\vee})$ in the face of it that lies above $\Lambda^{\ast}$ . The inverse image of this face agrees with $\Delta_{\psi,\Lambda}$ .

Since we have an explicit description of the polytope $\Delta_{\psi,\Lambda}$ , we can easily calculate the degree with respect to $D_{\psi}$ of an orbit $V(\Lambda)$ .

Proposition 4.105.

Let $\Pi$ be a complete SCR polyhedral complex in $N_{\mathbb{R}}$ and $\psi$ an H-lattice concave function on $\Pi$ . Let $\Lambda\in\Pi$ be a polyhedron of dimension $n-k$ , $u_{0}\in\operatorname{ri}(\Lambda)$ and $\Lambda^{\ast}=\partial\psi(u_{0})$ . Then

(4.106)

\operatorname{mult}(\Lambda)\deg_{D_{\psi}}(V(\Lambda))=k!\operatorname{vol}_{M(\Lambda)}(\Lambda^{\ast}),

where $\operatorname{mult}(\Lambda)$ is the multiplicity of $\Lambda$ (see Definition 4.68).

Proof.

From the description of $D_{\psi}|_{V(\Lambda)}$ and Proposition 4.37, we know that

\deg_{D_{\psi}}(V(\Lambda))=k!\operatorname{vol}_{{\widetilde{M}}(\Lambda)}(\Delta_{\psi,\Lambda}).

Since

\operatorname{vol}_{{\widetilde{M}}(\Lambda)}(\Delta_{\psi,\Lambda})=\frac{1}{[M(\Lambda):{\widetilde{M}}(\Lambda)]}\operatorname{vol}_{M(\Lambda)}(\Lambda^{\ast}),

the result follows from the definition of the multiplicity. ∎

Remark 4.107.

If $\dim(\Lambda^{\ast})<k$ , then both sides of (4.106) are zero. If $\dim(\Lambda^{\ast})=k$ , then $M(\Lambda)=M(\Lambda^{\ast})$ and $\operatorname{vol}_{M(\Lambda)}(\Lambda^{\ast})$ agrees with the lattice volume of $\Lambda^{\ast}$ .

We now interpret the inverse image of a semipositive $\mathbb{T}$ -Cartier divisor by an equivariant morphism in terms of direct and inverse images of concave functions.

Proposition 4.108.

With the hypothesis of Proposition 4.72, let $\psi_{2}$ be an H-lattice concave function on $\Pi_{2}$ and let $D_{\psi_{2}}$ be the corresponding semipositive $\mathbb{T}$ -Cartier divisor. Then $\Phi_{p,A}^{\ast}D_{\psi_{2}}$ is the semipositive $\mathbb{T}$ -Cartier divisor associated to the H-lattice concave function $\psi_{1}=A^{\ast}\psi_{2}$ . Moreover the Legendre-Fenchel dual is given by

\psi_{1}^{\vee}=(H^{\vee})_{\ast}(\psi_{2}^{\vee}-{\operatorname{val}}(p)).

Proof.

The first statement is Proposition 4.94. The second statement follows from Proposition 3.78(1). ∎

Example 4.109.

Let $\Sigma$ be a complete fan in $N_{\mathbb{R}}$ and $\Psi$ a support function on $\Sigma$ . By Theorem 4.97, any equivalence class of semipositive models of $(X_{\Sigma},D_{\Psi})$ is determined by a rational piecewise affine concave function $\psi$ with $\operatorname{rec}(\psi)=\Psi$ . By Lemma 3.79, any such function can be realized as the inverse image by an affine map of the support function of a standard simplex. Using the previous proposition, any equivalence class of semipositive toric models can be induced by an equivariant projective morphism.

More explicitly, let $e>0$ be an integer such that $e\psi$ is an H-lattice concave function. Let $\Pi$ be a complete SCR complex in $N_{\mathbb{R}}$ compatible by $e\psi$ and such that $\operatorname{rec}(\Pi)=\Sigma$ (see the proof of Theorem 4.97). Then, $({\mathcal{X}}_{\Pi},D_{e\psi},e)$ is a toric model of $(X_{\Sigma},D_{\Psi})$ in the class determined by $\psi$ .

Choose an H-representation $e\psi(u)=\min_{0\leq i\leq r}(m_{i}(u)+l_{i})$ with $(m_{i},l_{i})\in{\widetilde{M}}$ for $i=0,\dots,r$ . Put $\boldsymbol{\alpha}=(l_{1}-l_{0},\dots,l_{r}-l_{0})$ . Let $H$ and $A$ be as in Lemma 3.79. In our case, $H$ is a morphism of lattices and

(4.110)

e\psi=A^{\ast}\Psi_{\Delta^{r}}+m_{0}+l_{0}.

We follow examples 4.3, 4.26, 4.44 and 4.75, and consider $\mathbb{P}^{r}_{S}$ as a toric scheme over $S$ . Let $p=(p_{0}:\dots:p_{r})$ be a rational point in the principal open subset of $\mathbb{P}^{r}_{K}$ such that ${\operatorname{val}}(p)=\boldsymbol{\alpha}$ . One can verify that the hypothesis of Proposition 4.72 are satisfied. Let $\Phi_{p,A}\colon{\mathcal{X}}_{\Pi}\to\mathbb{P}^{r}_{S}$ be the associated morphism. Then

D_{e\psi}=\Phi_{p,A}^{\ast}D_{\Psi_{\Delta^{r}}}+\operatorname{div}(\varpi^{-l_{0}}\chi^{-m_{0}}).