\[rem:reflexive\] If $\D\subseteq M_\R$ is a reflexive polytope, then $$\P_\D\cong\XX_{\D^*}\quad\text{and}\quad\XX_\D\cong\P_{\D^*}$$ These isomorphisms are induced by identity morphisms of lattices $N$ and $M$, respectively.

\[cor:smallres\] Let $\D$ be a lattice polytope such that $\0\in\operatorname{Int}(\D)$ and $V_\D$ be the fan matrix of $\XX_\D$, as constructed in the previous Proposition \[prop:Fmatricedipolitopo\]. Then, for any $\Si\in\SF(V_\D)$ which is a refinement of $\Si_\D$, the associated toric variety $X(\Si)$ is a $\Q$-factorial small resolution of $\XX_\D$.

In fact, every refinement $\Si\in\SF(V)$ of $\Si_\D$ is obtained by a simplicial subdivision of cones in $\Si_\D$. In particular, the induced birational resolution $X(\Si)\longrightarrow\XX_\D$ is small, as $\Si(1)=\Si_\D(1)=\{\langle\v_1\rangle,\ldots,\langle\v_m\rangle\}$.

Cones of divisors {#ssez:coni&div}
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Let $X(\Si)$ be a complete toric variety. Then there is a short exact sequence $$\label{complete deg sequence}
  \xymatrix@1{0\ar[r]& M \ar[r]^-{div}_-{V^T} & *!U(.45){\bigoplus\limits_{\rho \in \Sigma (1)} \Z \cdot D_{\rho}}
\ar[r]^-d_-Q & \operatorname{Cl}(X) \ar[r] & 0 }$$ (see e.g. [@Fulton §3.4], [@CLS Prop. 4.2.5]). The representative matrices, $V$ and $Q$, of the $\Z$-linear morphisms $div$ and $d$, respectively, gives a fan matrix and *weight matrix*, respectively, of $X$. Since $X$ is complete, $V$ is a reduced $F$-matrix and $Q$ is a *Gale dual matrix* of $V$, which *can be assumed to be positive* . This means that:

-   *the 