The special fibre of $\mathcal{X}_{\Pi}$ is a chain of rational curves $E_{i}$, $i=0,\dots,k$, corresponding to the points $a-i$. The monomial $\chi^{1}$ is a section of the trivial line bundle and corresponds to the function $\psi(u)=-u$. Using Proposition 4.84 we obtain that

where $D_{0}$ and $D_{\infty}$ are again the horizontal divisors determined by the points $0$ and $\infty$.

Since the vertices of the polyhedral complex $\Pi$ are integral, by equation (4.87), we deduce that $\operatorname{div}(\varpi)$ is reduced.

Then the result follows from [Lic68, Corollary 1.13] using an explicit description of the local rings at the points of the special fibre as in the proof of Lemma 5.57. ∎