Let $X$ be a smooth projective surface. Let $C\subset X$ is a smooth and irreducible curve. By the Kawamata covering lemma, for any natural $m$ there exists a smooth projective surface $Y$ and a finite map $p:Y\rightarrow X$ together with a smooth divisor $D\subset Y$ such that $p^*C= mD$. Now will $D$ be irreducible. Will it be a disjoint union of smooth curves, if so are they isomorphic? Can choose $Y$ such that $D$ is irreducible?