Let $\Sigma$ be a complete fan in $N_{\mathbb{R}}$ and $\Psi$ a support function on $\Sigma$. By Theorem 4.97, any equivalence class of semipositive models of $(X_{\Sigma},D_{\Psi})$ is determined by a rational piecewise affine concave function $\psi$ with $\operatorname{rec}(\psi)=\Psi$. By Lemma 3.79, any such function can be realized as the inverse image by an affine map of the support function of a standard simplex. Using the previous proposition, any equivalence class of semipositive toric models can be induced by an equivariant projective morphism.

More explicitly, let $e>0$ be an integer such that $e\psi$ is an H-lattice concave function. Let $\Pi$ be a complete SCR complex in $N_{\mathbb{R}}$ compatible by $e\psi$ and such that $\operatorname{rec}(\Pi)=\Sigma$ (see the proof of Theorem 4.97). Then, $({\mathcal{X}}_{\Pi},D_{e\psi},e)$ is a toric model of $(X_{\Sigma},D_{\Psi})$ in the class determined by $\psi$.

Choose an H-representation $e\psi(u)=\min_{0\leq i\leq r}(m_{i}(u)+l_{i})$ with $(m_{i},l_{i})\in{\widetilde{M}}$ for $i=0,\dots,r$. Put $\boldsymbol{\alpha}=(l_{1}-l_{0},\dots,l_{r}-l_{0})$. Let $H$ and $A$ be as in Lemma 3.79. In our case, $H$ is a morphism of lattices and

We follow examples 4.3, 4.26, 4.44 and 4.75, and consider $\mathbb{P}^{r}_{S}$ as a toric scheme over $S$. Let $p=(p_{0}:\dots:p_{r})$ be a rational point in the principal open subset of $\mathbb{P}^{r}_{K}$ such that ${\operatorname{val}}(p)=\boldsymbol{\alpha}$. One can verify that the hypothesis of Proposition 4.72 are satisfied. Let $\Phi_{p,A}\colon{\mathcal{X}}_{\Pi}\to\mathbb{P}^{r}_{S}$ be the associated morphism. Then