Let $X$ be a non-singular projective variety over the field of complex numbers, of dimension $\geq 2$. Suppose $D$ is a non-singular and irreducible divisor of $X$.
The Kawamata covering lemma (Proposition 4.1.12, Positivity in Algebraic Geometry) says that - for any $m>0$, there is a smooth variety $Y$ and a finite flat morphism $f:Y\rightarrow X$ such that $f^*D=mD'$ for some smooth divisor $D'$ in $Y$.
(a) Looking at the proof, it seems to me that the divisor $D'$ is irreducible as well. Is this right?
(b) Is $f$ a Galois morphism? I think it is, since it is constructed from the Bloch-Gieseker covering and cyclic covering.
(c) Consider the morphism between non-singular irreducible varieties $f|_{D'}:D'\rightarrow D$. Can we say what the degree of this map is? Is it possible to obtain a covering $f$ so that $f|_{D'}$ is of some prescribed degree?