(2.1.4) For every $\mathscr{C}$-model $\mathscr{X}$ of $X$, we denote by $\mathscr{X}^{\mathrm{snc}}$ the subset of $\mathscr{X}$ consisting of the points where $\mathscr{X}$ is regular and $\mathscr{X}_{s}$ is a divisor with strict normal crossings (some authors use the terminology “simple normal crossings” instead). Thus $\mathscr{X}^{\mathrm{snc}}$ is the union of $X$ with the set of points $x$ of $\mathscr{X}_{s}$ such that $\mathcal{O}_{\mathscr{X},x}$ is regular and there exist a unit $u$ and a regular system of local parameters $(z_{1},\ldots,z_{n})$ in $\mathcal{O}_{\mathscr{X},x}$ and non-negative integers $N_{1},\ldots,N_{n}$ such that

$$t=u\prod_{i=1}^{n}(z_{i})^{N_{i}}.$$

The subset $\mathscr{X}^{\mathrm{snc}}$ is an open subscheme of $\mathscr{X}$ and it is again a $\mathscr{C}$-model of $X$. Moreover, if $\mathscr{X}$ is normal, then $\mathscr{X}^{\mathrm{snc}}_{s}$ is dense in $\mathscr{X}_{s}$. We say that $\mathscr{X}$ is an $snc$-model of $X$ if $\mathscr{X}=\mathscr{X}^{\mathrm{snc}}$, that is, if $\mathscr{X}$ is regular and $\mathscr{X}_{s}$ is a divisor with strict normal crossings. If $\mathscr{X}$ is a model of $X$ over $\mathscr{C}$, then a log resolution of $(\mathscr{X},\mathscr{X}_{s})$ is a proper morphism of $\mathscr{C}$-models $h:\mathscr{Y}\to\mathscr{X}$ such that $\mathscr{Y}$ is an $snc$-model of $X$.