The projective morphisms associated to $\mathbb{T}$-Cartier divisors generated by global sections can also be made explicit in terms of the lattice points of the associated polytope. Consider a complete toric variety $X_{\Sigma}$ of dimension $n$ equipped with a $\mathbb{T}$-Cartier divisor $D_{\Psi}$ generated by global sections. Let $m_{0},\dots,m_{r}\in\Delta_{\Psi}\cap M$ be such that $\operatorname{conv}(m_{0},\dots,m_{r})=\Delta_{\Psi}$. These vectors determine an H-representation $\Psi=\min_{i=0,\dots,r}m_{i}$. Let $H\colon N_{\mathbb{R}}\to\mathbb{R}^{r}$ be the linear map defined by $H(u)=(m_{i}(u)-m_{0}(u))_{i=1,\dots,r}$. By Lemma 3.79, $\Psi=H^{\ast}\Psi_{\Delta^{r}}+m_{0}$.

In $\mathbb{R}^{r}$ we consider the fan $\Sigma_{\Delta^{r}}$, whose associated toric variety is $\mathbb{P}^{r}$. One easily verifies that, for each $\sigma\in\Sigma$, there is $\sigma^{\prime}\in\Sigma_{\Delta^{r}}$ with $H(\sigma)\subset\sigma^{\prime}$. Let $p=(p_{0}:\dots:p_{r})$ be an arbitrary rational point of the principal open subset of $\mathbb{P}^{r}$. The equivariant morphism $\varphi_{p,H}\colon X\to\mathbb{P}^{r}_{K}$ can be written explicitly as $(p_{0}\chi^{m_{0}}:\dots:p_{r}\chi^{m_{r}})$. Moreover, $D_{\Psi}=\varphi_{p,H}^{\ast}D_{\Psi_{\Delta^{r}}}+\operatorname{div}(\chi^{-m_{0}})$.