Let $x$ be a point of $\mathrm{Sk}(X_{K},\omega)$. If $\mathrm{red}_{\mathscr{X}}(x)$ is contained in $\mathscr{X}^{\mathrm{snc}}$, then $x$ lies in $\widehat{\mathscr{X}}_{\eta}$ and [MN13, 4.4.5] implies that $x$ must lie in $\mathrm{Sk}(\mathscr{X})$, since the restriction of $\mathrm{wt}_{\omega}$ to $\widehat{\mathscr{X}}_{\eta}$ can reach its minimal values only at points of $\mathrm{Sk}(\mathscr{X})$.

Now suppose that $\mathrm{red}_{\mathscr{X}}(x)$ is not contained in $\mathscr{X}^{\mathrm{snc}}$. We will deduce a contradiction with the assumption that $x$ belongs to $\mathrm{Sk}(X_{K},\omega)$. Let $E$ be an irreducible component of $\mathscr{X}^{\mathrm{snc}}_{s}$ whose closure contains $\mathrm{red}_{\mathscr{X}}(x)$, let $\xi$ be the generic point of $E$ and denote by $x^{\prime}$ the unique point in $\mathrm{red}_{\mathscr{X}}^{-1}(\xi)$. We will prove that $\mathrm{wt}_{\omega}(x^{\prime})<\mathrm{wt}_{\omega}(x)$. Then $x$ cannot belong to the locus $\mathrm{Sk}(X_{K},\omega)$ where $\mathrm{wt}_{\omega}$ reaches its minimal value. Note that, since $\mathscr{X}$ is $\mathbb{Q}$-factorial, we have

(3.3.3)

$$|f(x^{\prime})|\geq|f(x)|$$

for every element $f$ of the local ring of $\mathscr{X}$ at $x$.

Replacing $\omega$ by its $d$-fold tensor power $\omega^{\otimes d}$, with $d$ a positive integer, has no influence on the skeleton $\mathrm{Sk}(X_{K},\omega)$. Thus we may assume that the divisor

$$mK_{\mathscr{X}}+m(\mathscr{X}_{s})_{\mathrm{red}}$$

is Cartier on $\mathscr{X}$ and we denote by $\mathcal{L}$ the associated line bundle. We choose a local generator $\theta$ of $\mathcal{L}$ at the point $\mathrm{red}_{\mathscr{X}}(x)$. Note that the pullback of $\mathcal{L}$ to the regular locus $\mathscr{X}^{\mathrm{reg}}_{R}$ of $\mathscr{X}_{R}$ is isomorphic to

$$\omega_{\mathscr{X}^{\mathrm{reg}}_{R}/R}((\mathscr{X}^{\mathrm{reg}}_{s})_{\mathrm{red}})^{\otimes m}.$$

We fix such an isomorphism. Then we can view $\omega$ as a rational section of $\mathcal{L}$ and write $\omega=g\theta$ locally at $\mathrm{red}_{\mathscr{X}}(x)$, with $g$ an element of

$$\mathcal{O}_{\mathscr{X}_{R},\mathrm{red}_{\mathscr{X}}(x)}\otimes_{R}K.$$

Then $\mathrm{wt}_{\omega}(x^{\prime})=-\ln|g(x^{\prime})|$. By (3.3.3), it is enough to show that

$$\mathrm{wt}_{\omega}(x)>-\ln|g(x)|.$$

Let $h:\mathscr{Y}\to\mathscr{X}$ be a log-resolution of $(\mathscr{X},\mathscr{X}_{s})$. Then $\mathrm{Sk}(X_{K},\omega)$ is contained in $\mathrm{Sk}(\mathscr{Y})$. We denote by $\Delta$ the log pullback of $(\mathscr{X}_{s})_{\mathrm{red}}$ to $\mathscr{Y}$. Locally at $\mathrm{red}_{\mathscr{Y}}(x)$, it is explicitly given by

$$\frac{1}{m}(\mathrm{div}(h^{*}g)-\mathrm{div}_{\mathscr{Y}}(\omega)).$$

Since $\mathscr{X}$ is a $dlt$-model and $\mathrm{red}_{\mathscr{X}}(x)$ does not belong to $\mathscr{X}^{\mathrm{snc}}$, we know that $\Delta<(\mathscr{Y}_{s})_{\mathrm{red}}$ locally around $\mathrm{red}_{\mathscr{Y}}(x)$ by Lemma 3.2.3. Therefore, we can write

	$\displaystyle\mathrm{wt}_{\omega}(x)$	$\displaystyle=$	$\displaystyle v_{x}(\mathrm{div}_{\mathscr{Y}}(\omega)+m(\mathscr{Y}_{s})_{\mathrm{red}})$
		$\displaystyle>$	$\displaystyle v_{x}(\mathrm{div}_{\mathscr{Y}}(\omega)+m\Delta)$
		$\displaystyle=$	$\displaystyle-\ln|g(x)|.$

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