S$ is a simple normal crossing divisor on $V$. If we choose $\Sigma$ as above, then we obtain that $\rho^*S$ is a simple normal crossing divisor on $V'$.

Finally, let us explain Viehweg’s fiber product trick. We include the proof for the benefit of the reader. We will use it in the proof of Theorems \[y-thm1.7\] and \[y-thm1.8\] in Section \[x-sec10\].

\[x-rem3.5\] Let $V$ be a reduced Gorenstein scheme. Note that $V$ may be reducible. We consider $$V'\overset{\delta}\longrightarrow V^\nu\overset{\nu}\longrightarrow V$$ where $\nu:V^\nu\to V$ is the normalization and $\delta:V'\to V^\nu$ is a resolution of singularities. Then, for every positive integer $n$, we have $$\label{x-3.1} 
\nu_*\mathcal O_{V^\nu}(nK_{V^\nu})\subset \omega^{\otimes n}_V$$ and $$\label{x-3.2}
\delta_*\mathcal O_{V'}(nK_{V'}+E)\subset \mathcal O_{V^\nu}(nK_{V^\nu})$$ where $E$ is any $\delta$-exceptional divisor on $V'$. In particular, we have $$\label{x-3.3}
(\nu\circ \delta)_*\mathcal O_{V'}(nK_{V'}+E)\subset 
\omega^{\otimes n}_V$$ for every positive integer $n$. If $U$ is a Zariski open set of $V$ such that $\nu\circ \delta$ is an isomorphism over $U$, then the inclusion is an isomorphism over $U$.

In Steps \[x-step3.5.1\] and \[x-step3.5.2\], we will prove and , respectively.

\[x-step3.5.1\] By taking the double dual of $\delta_*\mathcal O_{V'}(nK_{V'}+E)$, we obtain $\mathcal O_{V^\nu}(nK_{V^\nu})$. Therefore, we have $$\delta_*\mathcal O_{V'}(nK_{V'}+E)\subset \mathcal O_{V^\nu}(nK_{V^\nu})$$ for every integer $n$.

\[x-step3.5.2\] Since $\nu$ 