Let $X$ be a separated scheme of finite type over $K$. A separated flat $R$-scheme $\mathscr{X}$ of finite type together with an isomorphism of $K$-schemes $\mathscr{X}\times_{R}K\simeq X$ is called an $R$-model of $X$. We denote by $\mathscr{X}_{k}=\mathscr{X}\times_{R}k$ the special fiber of $\mathscr{X}$, and by $\Div_{0}(\mathscr{X})$ the group of Weil divisors on $\mathscr{X}$ supported on the special fiber.

If $Y$ is a normal variety and $D$ a Weil divisor on $Y$, whose irreducible decomposition is $D=\sum_{i\in I}a_{i}D_{i}$, a stratum of $D$ is a connected component of an intersection $D_{J}=\cap_{j\in J}D_{j}$ for some $J\subset I$. An open stratum of $D$ is a stratum $Z$ minus the irreducible components of $D$ not containing $Z$; this is denoted by $\mathring{Z}$.

We will not give a precise definition of log canonical centers here, and refer the reader to [KM98]. However, if $(\mathscr{X},\mathscr{X}_{k,\red})$ is dlt and defined over an algebraic curve - which is the most relevant case for applications - then the log canonical centers of $(\mathscr{X},\mathscr{X}_{k,\red})$ are precisely the strata of $\mathscr{X}_{k}$ by [Kol13, 4.16], so that a dlt model $\mathscr{X}$ is simple normal crossing at the generic points of the strata of $\mathscr{X}_{k}$. If $\mathscr{X}_{k}$ is reduced, it follows from the approximation arguments of [NXY19, Corollary 4.4] that this holds in the general case as well.

A dlt model $\mathscr{X}$ is good if each irreducible component of $\mathscr{X}_{k,\red}$ is $\mathbb{Q}$-Cartier. See [NXY19, §1.12-1.14] for an overview on existence results of such models.