Pome

Proposition 21.13.8111BC1BBBDE6

Proposition 21.13.8.111BC1BBBDE6 The base locus $\operatorname{Bs}(D)$ (with the reduced complex structure) of $D$ is a toric-invariant (possibly reducible) subvariety given by the union of $V(\tau )$, where $\tau $ runs over elements in $\Sigma $ satisfying the following condition:

0AA0B0B933FC 21.13.9
\begin{equation} a_{\rho }+\langle m, u_{\rho }\rangle >0\quad \textup{for some }\rho \in \tau (1) \end{equation}

for each $m\in M\cap P_ D$.

Proof. Recall that

\[ \mathrm{H}^0(X,L)\cong \bigoplus _{m\in M\cap P_ D} \mathbb {C}\chi ^ m. \]

See [Proposition 4.3.3, CLS11] for example. So we only need to understand the common zeros of $D+\operatorname{div}\chi ^ m$ for all $m\in M\cap P_ D$. But we know that

\[ D+\operatorname{div}\chi ^ m=\sum _{\rho \in \Sigma (1)} \left( a_{\rho }+\langle m, u_{\rho }\rangle \right) D_{\rho }. \]

Our assertion follows. $\square$

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