Sections of Cartier divisors on toric varieties

Question

Let $X_{\Sigma}$ be a projective toric variety. Consider the total coordinate ring $$S = \mathbb{C}[x_{\rho}\: | \: \rho\in\Sigma(1)]$$ Define $\deg(x_{\rho}) = D_{\rho}$. Now, take a divisor $D = \sum_{\rho\in\Sigma(1)}a_{\rho}D_{\rho}$. Then $H^0(X,D)$ is generated by monomials $X^{m}$ such that $m\in M$ with $<m,u_{\rho}>\geq -a_{\rho}$ for all $\rho\in\Sigma(1)$, where $u_{\rho}$ is the generator of the ray corresponding to $D_{\rho}$.

Take an $m\in M$ yielding a section of $D$. Is it then true that the monomial $x_{1}^{<m,u_1>+a_1}\dots x_{k}^{<m,u_k>+a_k}$, where $k = |\Sigma(1)|$, belongs to the degree $D$ part of $S$?

Answer 1

That is true. Just note that

$$deg(x_1^{<m,u_1>+a_1}\cdots x_k^{<m,u_k>+a_k})$$

is the class in $Cl(X_{\Sigma})$ of the divisor

$$\sum_{\rho} (<m,u_\rho>+a_\rho)D_\rho = \sum_{\rho}<m,u_\rho>D_\rho + \sum_{\rho}a_{\rho} D_\rho \sim div(\chi^m)+D\sim D$$