*integer part* of a polytope $\D\subseteq M_\R$ as $$[\D]:=\operatorname{Conv}(\{\m\in M\cap\D\})$$ Clearly, if $\D$ is a lattice polytope then $[\D]=\D$.

\[def:Deltapolytope\] The *framing polytope* (*$f$-polytope*) of a ftv $(X,\aa)$ is the lattice polytope $\D(X,\aa)\subseteq M_\R$ so defined: $$\label{k0}
     \D(X,\aa):=[k_0\D_\aa]\quad,\quad k_0:=\min\{k\in\N\,|\,\0\in\operatorname{Int}[k\D_\aa]\}$$

For what observed above, $k_0$ is well defined. Notice that $k_0$ may be bigger than 1: in fact it may happen that $\0$ is not an interior point of the integer part $[\D_\aa]$, as the following Example \[ex:k0&gt;1\] shows. On the other hand $k_0=1$ when $\D_\aa$ is a lattice polytope: in this case $\D(X,\aa)=\D_\aa$.

\[ex:k0&gt;1\] Consider the ftv given by $(X,\aa)=(\P(1,2,5),(2,1,1))$. Then $$\D_\aa=\operatorname{Conv}\left(
                   \begin{array}{ccc}
                     3 & {-3/ 5} & {-3/ 2} \\
                     1 & {-4/ 5} & 1 \\
                   \end{array}
                 \right)\ \Longrightarrow\ [\D_\aa]=\operatorname{Conv}\left(
                                                           \begin{array}{cccc}
                                                             3 & -1 & 1 & -1 \\
                                                             1 & 1 & 0 & 0 \\
                                                           \end{array}
                                                         \right)$$ Then $\0\not\in\operatorname{Int}[\D_\aa]$. But $\0\in\operatorname{Int}[2\D_\aa]$, so giving $k_0=2$ in the previous Definition \[def:Deltapolytope\].

\[rem:anticanonico\] Assume $\aa=\1$, that is $D_\aa=-K_X$ is the anti-canonical divisor of the complete toric variety $X$. Then, if $\D_\1$ is a reflexive polytope, the above construction gives $$\D(X,\aa)=\D_\aa=\D_\1=\D_{-K_X}$$

### The $f$-dual ftv and its small $\Q$-factorial resolutions {#sssez:deltadual}

Associated with the construction of the lattice polytope $\D(X,\aa)$ 