Proposition 21.9.3.C62B8871AA1C Let $v$ and $v'$ be vertices of $P$, with primitive edge bases $e_1,\ldots ,e_ n$ and $e_1',\ldots ,e_ n'$ and vertex charts $w$ and $w'$, and let $u_1,\ldots ,u_ n$ be the inward normals of the facets through $v$, as in lemma 21.8.2(4).
A lattice point $m\in M$ satisfies $\langle m,x\rangle \geq 0$ for all $x\in \sigma _{\{ v\} }$ if and only if $m=\sum _ jc_ je_ j$ with integers $c_ j\geq 0$. Consequently, the characters that are holomorphic on $U_ v$ are exactly the monomials $w^ c=\prod _ jw_ j^{c_ j}$, $c\in \mathbb Z_{\geq 0}^ n$.
The character $\chi ^{v'-v}$ is holomorphic on $U_ v$, and $U_ v\cap U_{v'}$ is the set of points of $U_ v$ at which it does not vanish. On this set
\[ w'_ i=\prod _ jw_ j^{\langle e'_ i,u_ j\rangle }\qquad (1\leq i\leq n), \]
where the factors with negative exponents do not vanish.
Let $Q$ be the intersection of $P$ with the facets containing both $v$ and $v'$, the smallest face of $P$ containing $v$ and $v'$. Then
\[ \sigma _{\{ v\} }\cap \sigma _{\{ v'\} }=\sigma _ Q=\{ x\in \sigma _{\{ v\} }:\langle v'-v,x\rangle =0\} . \]
Proof.
(1) Since $u_1,\ldots ,u_ n$ is the basis of $N$ dual to $e_1,\ldots ,e_ n$, every $m\in M$ is $m=\sum _ j\langle m,u_ j\rangle e_ j$ with integer coefficients, and $\sigma _{\{ v\} }=\operatorname {cone}(u_1,\ldots ,u_ n)$. Hence $\langle m,\cdot \rangle \geq 0$ on $\sigma _{\{ v\} }$ if and only if all coefficients $\langle m,u_ j\rangle $ are nonnegative. The second assertion follows from the last assertion of proposition 21.9.2, since $\chi ^ m=\prod _ jw_ j^{\langle m,u_ j\rangle }$ on $U_ v$.
(2) By theorem 21.8.3, $Z_{v'}/Z_ v=\prod _ jw_ j^{b_ j(v')}$ on $U_ v$, with $b_ j(v')=\ell _ j(v')=\langle v'-v,u_ j\rangle \geq 0$; this holomorphic function is the extension of $\chi ^{v'-v}$ to $U_ v$. Since $U_ v\cap U_{v'}=U_ v\cap \{ Z_{v'}\neq 0\} $, the first assertion follows. On $U_ v\cap U_{v'}$ we have $w'_ i=Z_{v'+e_ i'}/Z_{v'}=w^{b(v'+e_ i')}/w^{b(v')}$, a quotient of monomials whose denominator does not vanish there, and $b_ j(v'+e'_ i)-b_ j(v')=\langle e'_ i,u_ j\rangle $. A negative exponent $\langle e'_ i,u_ j\rangle $ forces $b_ j(v')>0$, so the corresponding factor $w_ j$ divides the denominator and does not vanish on $U_ v\cap U_{v'}$.
(3) By proposition 21.9.1, applied to the faces $\{ v\} $ and $\{ v'\} $, the set $Q$ is the smallest face containing $v$ and $v'$, and $\sigma _{\{ v\} }\cap \sigma _{\{ v'\} }=\sigma _ Q$. The facets containing $Q$ are the facets $\rho _ j$ through $v$ with $\ell _ j(v')=0$. If $x=\sum _ j\alpha _ ju_ j\in \sigma _{\{ v\} }$ with $\alpha _ j\geq 0$, then $\langle v'-v,x\rangle =\sum _ j\alpha _ j\ell _ j(v')$ is a sum of nonnegative terms, so it vanishes if and only if $\alpha _ j=0$ whenever $\ell _ j(v')>0$, that is, if and only if $x\in \operatorname {cone}\{ u_ j:\ell _ j(v')=0\} =\sigma _ Q$.
$\square$
Pome
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