present section by introducing the main character of this paper.

\[def:ftv\] A *framed toric variety* is a couple $(X,D)$ where:

-   $X$ is a complete toric variety, with $\dim(X)=n$ and $\operatorname{rk}(\operatorname{Pic}(X))=r$,

-   $D=\sum_{\rho\in\Si(1)}a_\rho D_\rho=\sum_{i=1}^m a_i D_i\in\Weil(X)$, with $m=n+r$, is a *strictly effective* (that is $a_i>0$, for every $i$), torus invariant Weil divisor, called a *framing* of $X$.

A *morphism of framed toric varieties* $f:(X,D)\longrightarrow(X',D')$ is a morphism of underlying toric varieties $f:X\longrightarrow X'$ inducing a well defined pull-back morphism on torus invariant Weil divisors $f^*:\Weil(X')\longrightarrow\Weil(X)$ such that $f^*D'=D$. If $f$ is an isomorphism of toric varieties, then it gives an *isomorphism of framed toric varieties* $f:(X,D)\cong(X',D')$. It is well defined the category **ftv** of framed toric varieties.

Framed duality
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Given a ftv $$\left(X(\Si),D_\aa=\sum_{\rho\in\Si(1)} a_\rho D_\rho\right)=:(X,\aa)$$ consider the polytope associated with $D_\aa$ $$\label{Delta_a}
   \D_{\aa}:=\D_{D_\aa}=\{\m\in M_\R\,|\,V^T\cdot\m\geq -\aa\}$$ being $V$ a fan matrix of $X$. In general, $\D_\aa$ is the convex hull of a finite subset $\{\m_1,\ldots,\m_s\}\subset M_\Q:=M\otimes\Q$. Let $k\in\N$ be the minimum positive integer such that $k\m_i\in M$, for every $i=1,\ldots,s$. Then $$k\D_\aa=\D_{{kD_\aa}}=\operatorname{Conv}(k\m_1,\ldots,k\m_s)$$ is a lattice polytope. Since ${D_\aa}$ is strictly effective, that is $-\aa<\0$, certainly $\0\in\operatorname{Int}\D_\aa$, meaning that $\0\in\operatorname{Int}(k\D_\aa)$, for any positive integer $k\in\N$.

On the other hand, define the 