Pome

Proposition 21.13.330ED70BB2A7D0

Proposition 21.13.33.0ED70BB2A7D0 We have

35302703CBCC 21.13.34
\begin{equation} k^{-1}\nu _{Y_{\bullet }}\left(\mathrm{H}^0(X,L^ k)^{\times } \right)=\Phi _{\mathbb {R}} \left(P_{D}\cap k^{-1}M\right) \end{equation}

for any $k\in \mathbb {Z}_{>0}$. In particular,

12.28
\begin{equation} \Delta _{Y_{\bullet }}(L)=\Phi _{\mathbb {R}}(P_ D). \end{equation}

Proof. We first reduce to the case where $D|_{U_{\sigma }}=0$. In fact, replacing $D$ by $D+\operatorname{div}\chi ^{m_{\sigma }}$ would result in changing $P_ D$ to $P_ D-m_{\sigma }$. So in view of (21.13.6), we may assume that $D|_{U_{\sigma }}=0$ and hence $m_{\sigma }=0$.

Fix $k\in \mathbb {Z}_{>0}$. Let $s\in \mathrm{H}^0(X,L^ k)$ be a non-zero toric eigen-section, say $\chi ^ m$ for some $m\in kP_ D\cap M$. The zero-divisor of $s$ on $U_{\sigma }$ is given by

\[ \sum _{i=1}^ n \langle m, u_{\rho _ i} \rangle D_{\rho _ i}, \]

see [Proposition 4.1.2, CLS11]. Therefore,

\[ \nu _{Y_{\bullet }}(s)=\Bigl (\langle m, u_{\rho _1} \rangle ,\dots ,\langle m, u_{\rho _ n} \rangle \Bigr )=\Phi (m). \]

Conversely, every section of $L^ k$ is a finite linear combination of such characters. In the coordinates on $U_{\sigma }$ determined by $u_{\rho _1},\ldots ,u_{\rho _ n}$, the Okounkov valuation of a non-zero linear combination is the lexicographically first exponent among the characters with non-zero coefficient. Hence arbitrary sections give no valuations beyond the right-hand side of (21.13.34), and (21.13.34) follows. $\square$

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