Consider the constant H-lattice function $\psi_{\varpi}(u)=-1$. This function corresponds to the principal divisor $\operatorname{div}(\varpi)$. Then

(4.87)

$$\operatorname{div}(\varpi)=\sum_{v\in\Pi^{0}}\operatorname{mult}(v)V(v).$$

Thus, for a vertex $v$, the multiplicity of $v$ agrees with the multiplicity of the divisor $V(v)$ in the special fibre $\operatorname{div}(\varpi)$. In particular, the special fibre ${\mathcal{X}}_{\Pi,o}$ is reduced if and only if all vertexes of $\Pi^{0}$ belong to $N$.