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 $Z$ over the complex numbers and a reduced effective divisors with simple normal crossings $D$. Let $V=Z-D$. Let $U \to V$ be an étale cover.
What's the meaning of taking $\pi: Y \to X$ the normalization of $Z$ in the function field $\mathbb{C}(U)$?
Does it mean that $U \hookrightarrow Y$ and that the complement is a divisor with normal crossings laying above $D$?
Thanks for your help