Writing $\mathcal{L}=(\mathcal{A}+\mathcal{L})-\mathcal{A}$ for some sufficiently ample $\mathcal{A}\in\Pic(\mathcal{X})$ reduces us to the case where $\mathcal{L}$ is very ample. We then get a closed embedding $i:\mathcal{X}\hookrightarrow\mathbf{P}^{N}_{k}\times_{k}S$ over $S$ such that $\mathcal{L}$ coincides with the restriction of $\mathcal{O}(1)$. Let $\pi:\mathbf{P}^{N}_{k}\to\mathbf{P}^{N}_{k}$ be the morphism $[X_{0}:\dots:X_{N}]\mapsto[X_{0}^{m}:\dots:X_{N}^{m}]$, which satisfies $\pi^{*}\mathcal{O}(1)=\mathcal{O}(m)$. For each $\sigma\in\mathrm{PGL}(N+1,k)$ set $i_{\sigma}:=\sigma\circ i$ and consider $\mathcal{X}^{\prime}:=\mathcal{X}\times_{i_{\sigma}}\pi$ with the finite surjective morphism $\rho:\mathcal{X}^{\prime}\to\mathcal{X}$, so that $\rho^{*}\mathcal{L}=\mathcal{O}(m)|_{\mathcal{Y}}$ is divisible by $m$ in $\Pic(\mathcal{Y})$.

Applying Kleiman’s Bertini-type theorem (cf. [Har77, III.10.8]) to the smooth $k$-varieties $E_{J}=\bigcap_{j\in J}E_{j}$ for all subsets $J\subset I$ shows that we may choose $\sigma\in\mathrm{PGL}(N+1,k)$ such that each $\rho^{*}E_{i}$ is smooth over $k$ and $\sum_{i}\rho^{*}E_{i}$ has simple normal crossings. This implies in particular that $\mathcal{X}^{\prime}$ is an SNC model. ∎

As in Theorem B.1 the desired result is equivalent to $H^{q}(\mathcal{X},\mathcal{L}^{-1})=0$ for $q<n$ by relative duality. By flat base change we may assume that $k$ is algebraically closed.

Since $\rho^{*}{E_{i}}$ is smooth for each $i$ and $\sum_{i=2}^{N}\rho^{*}E_{i}$ has normal crossings with $\mathrm{div}(s)=\rho^{*}E_{1}$, one sees as in [KM98, Claim 2.65] that $E_{i}^{(1)}:=\tau^{*}\rho^{*}E_{i}$ is smooth for each $i$ and $\mathcal{X}_{1,0}$ has SNC support, so that $\mathcal{X}_{1}$ is an SNC $S$-scheme. Finally $\mathcal{L}_{1}-\sum_{i=2}^{N}a_{i}E_{i}^{(1)}$ is $\mathbf{Q}$-linearly equivalent to $\tau^{*}\rho^{*}(\mathcal{L}-D)$, hence is ample.

We now use Lemma B.2 to find $c_{1}:\mathcal{X}_{1}^{\prime}\to\mathcal{X}_{1}$ such that $c_{1}^{*}E_{i}^{(1)}$ is smooth for all $i$, $\sum_{i}c_{1}^{*}E_{i}^{(1)}$ is SNC and $c_{1}^{*}E_{2}^{(1)}$ is divisible in $\Pic(\mathcal{X}_{1}^{\prime})$ by the denominator of $a_{2}$. We then perform the same cyclic cover construction as above. Iterating the whole process finally yields an SNC $S$-variety $\mathcal{X}_{N}$ with an ample line bundle $\mathcal{L}_{N}$ such that $H^{q}(\mathcal{X},\mathcal{L}^{-1})$ is a direct summand of $H^{q}(\mathcal{X}_{N},\mathcal{L}_{N}^{-1})$, and we conclude by Theorem B.1.

We are next going to show that $R^{q}\pi_{*}(\omega_{\mathcal{Y}}\otimes\pi^{*}\mathcal{L})=0$ for each $q\geq 1$. Since we have $\pi_{*}\omega_{\mathcal{Y}}=\omega_{\mathcal{X}}$ (the relative canonical bundle $K_{\mathcal{Y}/\mathcal{X}}$ is $\pi$-exceptional and effective since $\mathcal{X}$ is regular), the degeneration of the Leray spectral sequence of $\pi$ will then yield as desired

for $q\geq 1$. Let us now prove the claim. Given $q\geq 1$ choose $\mathcal{A}\in\Pic(\mathcal{X})$ sufficiently ample to guarantee that $\mathcal{A}\otimes R^{q}\pi_{*}(\omega_{\mathcal{Y}}\otimes\pi^{*}\mathcal{L})$ is globally generated on $\mathcal{X}$ and

(note that we are only imposing finitely many non-trivial conditions). The degeneration of the Leray spectral sequence yields

for $q\geq 1$ by Step 1 again, since $\pi^{*}(\mathcal{L}\otimes\mathcal{A})-\varepsilon E$ is also ample. It follows that $\mathcal{A}\otimes R^{q}\pi_{*}(\omega_{\mathcal{Y}}\otimes\pi^{*}\mathcal{L})=0$ by global generation, which proves the claim since $\mathcal{A}$ is invertible. ∎