which is the closure of the one generated by classes of *movable* divisors, and the cone $\operatorname{Nef}(X)$, generated by classes of nef divisors, are both sub-cones of the effective cone $\overline{\operatorname{Eff}}(X)$. Then, the isomorphism (\[iso-coni\]) descends to give isomorphisms $$\overline{\operatorname{Mov}}(X)\cong\operatorname{Mov}(Q)\quad,\quad \operatorname{Nef}(X)\cong\operatorname{Nef}(\I_\Si)$$ More precisely, recalling notation \[sssez:SF\], we get the following

*[@CLS Thm. 15.1.10(c)]*\[prop:nef\] If $V=\left(
                                                                              \begin{array}{ccc}
                                                                                \v_1 & \ldots & \v_{n+r} \\
                                                                              \end{array}
                                                                            \right)$ is an $F$–matrix then, for every fan $\Si\in\P\SF(V)$ there is a natural isomorphism $$\operatorname{Pic}(X(\Si))\otimes\R\cong\operatorname{Cl}(X)\otimes\R\cong \R^r$$ taking the cones $$\operatorname{Nef}(X(\Si))\subseteq\overline{\operatorname{Mov}}(X(\Si))\subseteq\overline{\operatorname{Eff}}(X(\Si))\subseteq\R^r$$ to the sub-cones of the positive orthant $$\operatorname{Nef}(\I_\Si)\subseteq\operatorname{Mov}(Q)\subseteq\langle Q\rangle\subseteq\R^r_+$$ In particular, if $d:\mathcal{W}_T(X(\Si))\to\operatorname{Cl}(X(\Si))$ is the degree morphism, then a Weil divisor $D$ on $X(\Si)$ admits a nef (ample) positive multiple if and only if its class $[D]=d(D)\in\operatorname{Nef}(\I_\Si)$ ($d(D)\in\operatorname{Relint}\operatorname{Nef}(\I_\Si)$, resp.).

The following is a useful application to $\Q$-factorial small resolutions of a complete toric variety $X(\Si)$.

\[prop:risoluzioni\]

Let $X(\Si)$ be a complete toric variety and $V$ be a fan matrix of $X$. Then, for any $\Xi\in\SF(V)$ giving a refinement of $\Si$, the identity morphism of the common lattice $N$, namely $\operatorname{id}_N:N\longrightarrow N$, induces a fans’ morphism from $\Xi$ to $\Si$. The induced morphism of toric varieties $$\vf:Y(\Xi)\longrightarrow X(\Si)$$ is a $\Q$-factorial small resolution. Then, there is the following 