Topology of complex variety $Y^2=F(X)$

Question

Suppose $F\in\mathbb C[X]$ is a polynomial of degree $n=2k$ where $k\ge2$ is an integer. Suppose further that the $n$ roots $\alpha_1,\ldots,\alpha_n$ are distinct. Then, consider the algebraic subset $S=\{(z,w)\in\mathbb C^2\;|\;w^2=P(z)\}$ of the plane $\mathbb C^2$. Is there a simple topological description of this set $S$ (such as a sphere with handles attached and some points removed)? Also, what is the topological description of the closure of $S$ in $\mathbb C\mathbb P^2$ (where we embed $\mathbb C^2$ in $\mathbb C\mathbb P^2$ as $(z,w)\mapsto[z:w:1]$)? My guess is that $S$ is a genus $k-1$ surface with two points removed and that $\bar S$ (the closure) is a genus $k$ surface with one of the homology generators shrunk to a point). Are these descriptions correct?