Let $\mathcal{X}$ be an $S$-variety $\mathcal{X}$. We say that $\mathcal{X}$ is vertically $\mathbf{Q}$-factorial if each component $E_{i}$ is $\mathbf{Q}$-Cartier. We say that $\mathcal{X}$ is SNC if:

• (i)

  the special fiber $\mathcal{X}_{0}$ has simple normal crossing support;

• (ii)

  $E_{J}$ is irreducible (or empty) for each $J\subset I$.