I have seen the following construction and I would be very happy if someone could explain its meaning to me.
We start from a smooth projective algebraic variety $X$ over a field of characteristic zero $k$ and a reduced effective divisor with simple normal crossings $D$. Let $V=Z−D$ and let $U \to V$ be an étale cover.
What's the meaning of taking $\pi: Y \to X$ the normalization of $X$ in the function field $k(U)$?
Does it mean that $U \hookrightarrow Y$ and that the complement is a divisor with normal crossings $E$ such that $\pi(E)=D$?
Thanks for your help