We refer to [dFKX12], [NX13, §2.2] and [KNX15] for more details and references on what follows.

Let ${\mathcal{X}}$ be a model with $K^{\mathrm{log}}_{\mathcal{X}}$ ${\mathbb{Q}}$-Cartier, and recall that ${\mathcal{X}}^{\mathrm{div}}$ denotes the set of divisorial valuations $v$ on ${\mathcal{X}}$ such that $v({t})=1$. We define the log discrepancy $A_{\mathcal{X}}(v)$ as the log discrepancy of $v$ with respect to the pair $({\mathcal{X}},{\mathcal{X}}_{0,\mathrm{red}})$, in the usual sense of the Minimal Model Program.

The log discrepancy function $A_{\mathcal{X}}\colon{\mathcal{X}}^{\mathrm{div}}\to{\mathbb{Q}}$ is characterized by the following property: if ${\mathcal{X}}^{\prime}$ is a model over ${\mathcal{X}}$ with proper birational morphism $\rho\colon{\mathcal{X}}^{\prime}\to{\mathcal{X}}$, then

$$K^{\mathrm{log}}_{{\mathcal{X}}^{\prime}}=\rho^{*}K^{\mathrm{log}}_{\mathcal{X}}+\sum_{E}b_{E}A_{\mathcal{X}}(v_{E})E,$$

(5.2)

with $E$ running over the irreducible components of ${\mathcal{X}}^{\prime}_{0}$.

We say that a model ${\mathcal{X}}$ is log canonical (lc for short), Kawamata log terminal (klt) or divisorially log terminal (dlt) if the pair $({\mathcal{X}},{\mathcal{X}}_{0,\mathrm{red}})$ has this property, in the sense of the Minimal Model Program.

Since the generic fiber $X$ is smooth, a model ${\mathcal{X}}$ is thus lc (resp. klt) if and only if $K^{\mathrm{log}}_{\mathcal{X}}$ is ${\mathbb{Q}}$-Cartier, with log discrepancy function $A_{\mathcal{X}}\colon{\mathcal{X}}^{\mathrm{div}}\to{\mathbb{Q}}$ taking non-negative (resp. positive) values. If ${\mathcal{X}}$ is lc, then the center $c_{\mathcal{X}}(v)\in{\mathcal{X}}_{0}$ of a valuation $v\in{\mathcal{X}}^{\mathrm{div}}$ with $A_{\mathcal{X}}(v)=0$ is called an lc center of ${\mathcal{X}}$, and an lc model ${\mathcal{X}}$ is dlt if and only if ${\mathcal{X}}_{\mathrm{snc}}$ contains all lc centers. The irreducible components of each non-empty $E_{J}$ are then normal, with generic point contained in ${\mathcal{X}}_{\mathrm{snc}}$ [Kol13, 4.16].