Let $\Pi$ and $\psi$ be as before and let $\Lambda\in\Pi$. Let $m_{\Lambda}\in M$ and $l_{\Lambda}\in\mathbb{Z}$ be such that $\psi|_{\Lambda}=(m_{\Lambda}+l_{\Lambda})|_{\Lambda}$. Let ${\widetilde{\pi}}_{\Lambda}\colon{\widetilde{N}}_{\mathbb{R}}\to{\widetilde{N}}(\Lambda)_{\mathbb{R}}$ be the projection, and ${\widetilde{\pi}}^{\vee}_{\Lambda}\colon{\widetilde{M}}(\Lambda)_{\mathbb{R}}\to{\widetilde{M}}_{\mathbb{R}}$ the dual map. Then

(4.104)

$$(\psi-m_{\Lambda}-l_{\Lambda})(\Lambda)=({\widetilde{\pi}}_{\Lambda})_{\ast}(\operatorname{c}(\psi-m_{\Lambda}-l_{\Lambda})).$$

Moreover, this is a support function on the fan $\Pi(\Lambda)$. Its stability set is the polytope $\Delta_{\psi,\Lambda}:=({\widetilde{\pi}}^{\vee}_{\Lambda}+(m_{\Lambda},l_{\Lambda}))^{-1}\operatorname{epi}(-\psi^{\vee})$. Hence, the restriction of the divisor $D_{\psi-m_{\Lambda}-l_{\Lambda}}$ to the variety $V(\Lambda)$ is the divisor associated to the support function of $\Delta_{\psi,\Lambda}$