21.12.1 Toric setupDD9C13434AA7

Let $T$ be a complex torus of dimension $n$¹ and let $T_ c\subseteq T(\mathbb {C})$ denote the corresponding compact torus.

Write $M$ for the character lattice of $T$, which is a free Abelian group of rank $n$. Similarly, let $N$ be the cocharacter lattice of $T$, which is the dual lattice of $M$. Given $m\in M$, the corresponding character of $T$ is denoted by $\chi ^ m$. Write $M_{\mathbb {R}}=M\otimes _{\mathbb {Z}} \mathbb {R}$ and $N_{\mathbb {R}}=N\otimes _{\mathbb {Z}}\mathbb {R}$. The pairing between $M_{\mathbb {R}}$ and $N_{\mathbb {R}}$ is denoted by $\langle \bullet ,\bullet \rangle $.

Let $P\subseteq M_{\mathbb {R}}$ be a full-dimensional smooth² lattice polytope³.

Given any (closed) facet $F$ of $P$, let $u_ F\in N$ denote the unique ray generator (the first non-zero integral element) of the inward normal ray of $F$. Then $P$ can be represented as

for some uniquely determined integers $a_ F$. The presentation is called the facet presentation of $P$.

Given any (closed) face $Q$ of $P$, we let $\sigma _ Q\subseteq N_{\mathbb {R}}$ be the closed convex cone generated by the $u_ F$'s, where $F$ runs over all facets of $P$ containing $Q$. When $Q=P$, $\sigma _ P$ is understood as $\{ 0\} $.

Let $\Sigma $ be the (inner) normal fan of $P$. Namely,

The notation $\Sigma (1)$ denotes the set of rays in $\Sigma $. Note that $\Sigma (1)$ is in bijective correspondence with the set of facets of $P$. In fact, given any facet $F$ of $P$, the cone $\sigma _ F$ is just the ray generated by $u_ F$, namely, the inward normal ray of $F$.

For any $\rho \in \Sigma (1)$, let $u_{\rho }\in N$ denote the ray generator of $\rho $, namely the first non-zero element in $N\cap \rho $. If $\rho =\sigma _ F$ for some facet $F$ of $P$, then $u_{\rho }=u_ F$.

Now the facet presentation (21.12.2) can be equivalently rewritten as

Let $\operatorname{Supp}_ P\colon N_{\mathbb {R}} \rightarrow \mathbb {R}$ denote the support function of $P$. Recall that the support function (example 21.11.6) of $P$ is defined as

The support function uses a maximum.

Recall that the characteristic function $\chi _ P\colon M_{\mathbb {R}}\rightarrow \{ 0,\infty \} $ of $P$ is defined as in example 21.11.5:

Let $X=X_{\Sigma }$ be the smooth projective toric variety corresponding to $\Sigma $. See [Theorem 3.1.5, CLS11] for the construction of $X$ and [Theorem 3.1.19, CLS11] for the smoothness of $X$. There is a canonical embedding $T\subseteq X$ as a dense Zariski open subset.

Let $D$ be the Cartier divisor on $X$ defined by $P$:

where $D_{\rho }$ is the toric prime divisor defined by $\rho $ under the orbit-cone correspondence Proposition 21.9.2.

Let $L$ be the toric line bundle induced by $P$, namely $L=\mathcal{O}_ X(D)$. Since $P$ is a smooth (Delzant) lattice polytope, $L$ is very ample by [Proposition 2.4.4, Proposition 6.1.10, CLS11]; in particular, $L$ is ample.

We will choose the base $\mathrm{e}$ for the logarithm map

This choice is fixed throughout these constructions. We have a canonical identification

The logarithm map then induces a tropicalization map after tensoring with $N$:

Example 21.12.4.4214559F0471 We take $n=1$ and $P=[0,1]\subseteq M_{\mathbb {R}}=\mathbb {R}$. In this case, the facet representation (21.12.2) becomes

\[ P=\bigl \{ m\in \mathbb {R}: \langle m,1 \rangle \geq 0, \langle m,-1 \rangle \geq -1\bigr \} , \]

with $u_{\{ 0\} }=1$, $u_{\{ 1\} }=-1$, $a_{\{ 0\} }=0$ and $a_{\{ 1\} }=1$. The normal fan $\Sigma $ is

\[ \Sigma =\bigl \{ (-\infty ,0], \{ 0\} , [0,\infty ) \bigr \} . \]

See 21.12.5.

The corresponding toric variety is just $X=\mathbb {P}^1$. Under the orbit-cone correspondence, we have

\[ D_{[0,\infty )}=[0],\quad D_{(-\infty ,0]}=[\infty ]. \]

The associated divisor $D=[\infty ]$ and therefore,

\[ L=\mathcal{O}_ X(D)=\mathcal{O}_{\mathbb {P}^1}(1). \]