\langle Q^J\rangle$$ Results on divisorial polytopes come, on the one hand, directly from the definition given in (\[div-politopo\]) and, on the other hand, from Proposition \[prop:semiampio\], giving that $[D]\in\operatorname{Nef}(X)$, and Proposition \[prop:gg\]. In fact, a positive multiple $kD$ is besepoint free, so giving that $k\D_D=\operatorname{Conv}(\{\m_\s\in M\,|\,\s\in\Si(n)\})$. Since $\operatorname{Nef}(X)\subseteq\operatorname{Nef}(Y)$, $[\vf^*(D)]=\overline{\vf}^*([D])\in\operatorname{Nef}(Y)$ so giving that $\vf^*(D)$ is semi-ample, too, still by Proposition \[prop:nef\]. Therefore $$k\D_D=k\D_{\vf^*(D)}=\operatorname{Conv}(\{\m_\tau\in M\,|\,\tau\in\Xi(n)\})=\operatorname{Conv}(\{\m_I\in M\,|\,I\in\I_{\Xi(n)}\})$$ where $\m_\tau=\m_I$ whenever $\tau=\langle V_I\rangle$ and $\m_I$ is defined as in (\[mI\]).

Non-degenerate hypersurfaces and their stratification {#ssez:regular}
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Given a toric variety $X=X(\Si)$, let $\T\subseteq X$ be the maximal acting torus on $X$. Consider a Laurent polynomial $$f=\sum_{\substack{m\in M\\\text{finite}}} c_m\chi^m\ ,\quad c_m\in\C^*$$ Denote by $Z_f\subseteq\T$ the zero-locus of $f$ in $\T$ and let $Y_f$ be its closure in $X$.

\[def:Si-regolare\] A Laurent polynomial $f$, and the associated hypersurfaces $Z_f\subseteq\T$ and $Y_f\subseteq X(\Si)$, are called *non-degenerate* $\Si$ (or, equivalently, *$\Si$-regular*) if, for every $\s\in\Si$, the associated $\s$-stratum $Y_{f,\s}:=Y_f\cap \T\cdot x_\s$ is empty or a smooth subvariety of codimension 1 in the torus orbit $\T\cdot x_\s$. In other words, non-degenerate means that $Y_f$ admits only *transversal intersections* with all the torus orbits $\T\cdot x_\s$, $\s\in\Si$.

A duality between framed toric varieties {#sez:dualita-ftv}
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Let us start the 