Suppose that there is a point $x\in\mathcal{X}_{0}$ which has a neighbourhood isomorphic to a neighbourhood of a dimension zero torus orbit of an affine Gorenstein toric variety $Y_{x}$. Such an affine variety is specified as follows. Set $M=\mathbb{Z}^{n}$, $M_{\mathbb{R}}=M\otimes_{\mathbb{Z}}\mathbb{R}$, $N=\operatorname{Hom}_{\mathbb{Z}}(M,\mathbb{Z})$, $N_{\mathbb{R}}=N\otimes_{\mathbb{Z}}\mathbb{R}$ with $n=\dim\mathcal{X}_{t}$. Then there is a lattice polytope $\sigma\subseteq M_{\mathbb{R}}$, $C(\sigma):=\{(rm,r)\,|\,m\in\sigma,r\geq 0\}\subseteq M_{\mathbb{R}}\oplus\mathbb{R}$, $P:={C(\sigma)}^{\scriptscriptstyle\vee}\cap(N\oplus\mathbb{Z})$ the monoid determined by the dual of the cone $C(\sigma)$, $Y_{x}=\operatorname{Spec}\mathbb{C}[P]$, and finally $f$ coincides with the monomial $z^{(0,1)}$.

Now let us take a small neighbourhood of $x$ of the form

$$\widetilde{U}_{\delta}=\{y\in\operatorname{Spec}\mathbb{C}[P]\,|\,\hbox{$|z^{p}|<\delta$ for all $p\in P$}\}.$$

This is an open set as the condition $|z^{p}|<\delta$ can be tested on a finite generating set for $P$, provided that $\delta<1$. Then show that for a given $t$, $|t|<1$ and $\epsilon=-2\pi/\log|t|$, if

$$\sigma_{t}:=\{m\in M_{\mathbb{R}}\,|\,\hbox{$\langle p,(m,1)\rangle>{\log\delta\over\log|t|}$ for all $p\in P$}\},$$

then

$$f^{-1}(t)\cap\widetilde{U}_{\delta}\cong X_{\epsilon}(\sigma_{t}).$$

Note that

$$\sigma:=\{m\in M_{\mathbb{R}}\,|\,\hbox{$\langle p,(m,1)\rangle\geq 0$ for all $p\in P$}\},$$

so $\sigma_{t}$ is an open subset of $\sigma$, and as $t\rightarrow 0$, $\sigma_{t}$ converges to the interior of $\sigma$. ∎