When $D_{\Psi}$ is only generated by its global sections, the polytope $\Delta_{\Psi}$ may not determine the variety $X_{\Sigma}$, but it does determine a polarized toric variety that is the image of $X_{\Sigma}$ by a toric morphism. Write $\Delta=\Delta_{\Psi}$ for short. Let $M(\Delta)$ be as in Notation 3.103 and choose $m\in\operatorname{aff}(\Delta)\cap M$. Set $N(\Delta)=M(\Delta)^{\vee}$. The translated polytope $\Delta-m$ has the same dimension as its ambient space $L_{\Delta}=M(\Delta)_{\mathbb{R}}$. By the theorem above, it defines a complete fan $\Sigma_{\Delta}$ in $N(\Delta)_{\mathbb{R}}$ together with a support function $\Psi_{\Delta}\colon N(\Delta)\to\mathbb{R}$. The projection $N\to N(\Delta)$ induces a toric morphism

$$\varphi\colon X_{\Sigma}\longrightarrow X_{\Sigma_{\Delta}},$$

the divisor $D_{\Psi_{\Delta}}$ is ample, and $D_{\Psi}=\varphi^{*}D_{\Psi_{\Delta}}+\operatorname{div}(\chi^{-m})$.