$$\left(Z^{(1)}_1\times _W \cdots \times _WZ^{(1)}_n\right)
\times _W \left(Z^{(2)}_1\times _W \cdots \times _WZ^{(2)}_n\right).$$ For the details, see, for example, [@esnault-viehweg 3.19. Lemma] and [@viehweg3 Lemma 2.5]. Let $S^{(i)}_j$ be the cyclic cover of $V$ associated to $\left(f^*\mathcal A\right)^{\otimes d} 
=\mathcal O_V(f^*D_i+f^*H^{(i)}_j)$. Then we define $V'$ as the normalization of $$\left(S^{(1)}_1\times _V \cdots \times _VS^{(1)}_n\right)
\times _V \left(S^{(2)}_1\times _V \cdots \times _VS^{(2)}_n\right).$$ Note that $\rho:V'\to V$ is a finite flat morphism between smooth quasi-projective varieties. Since $V\times _WW'\to V$ is finite and flat and $V$ is smooth, $V\times _W W'$ is Cohen–Macaulay (see, for example, [@kollar-mori Corollary 5.5]). By construction, we can easily see that $V\times _W W'$ is smooth in codimension one. Therefore, $V\times _W W'$ is normal. Since $\rho$ factors through $V\times _W W'$ by construction, we see that $V'=V\times _W W'$ by Zariski’s main theorem. By the above construction of $\tau:W'\to W$, we see that $\tau:W'\to W$ is Galois.

We give a very important remark on Lemma \[x-lem3.3\].

\[x-rem3.4\] In the proof of Lemma \[x-lem3.3\], let $S$ be any simple normal crossing divisor on $V$. Then we can choose the ramification locus $\Sigma$ of $\tau$ such that $f^*P\not\subset S$ for any irreducible component $P$ of $\Sigma$ and that $f^*\Sigma\cup 