# Plane algebraic curves in $\mathbb C^2$ are connected in the analytic topology.

Is there a "simple" proof, not involving much tools of Algebraic Geometry, to the fact that every irreducible affine curve $C=\{(z,w)\in\mathbb C^2\,:\, F(z,w)=0\}$ (where $F\in\mathbb C[X,Y]$ is irreducible) is connected in the analytical topology?

I know that the word simple is a bit vague, but the problem is the following: If one reads some introductory book about Riemann surfaces, where Riemann surfaces are defined by charts, a stadard example is to show that every irreducible non-singular affine curve is a connected Riemann Surface. Clearly the difficult point is to showing the connectedness, but this proof is often skipped, infact one read "It can be shown, with some algebraic geometry that..."

• Dear Galoisfan: perhaps in the statement you want $F$ to be irreducible. – Bruno Joyal Nov 27 '13 at 16:12
• Yes, I will edit – Dubious Nov 27 '13 at 16:16

Probably the best way to see this is by seeing $C$ as a ramified covering of $\mathbf C$, and showing that it is path-connected. Let $d$ be the degree (in the variable $w$) of $F(z, w)$.
Let $S$ be the set of values $z_0$ such that $F(z_0, w)$ has a multiple root. This is a finite subset of $\mathbf C$, determined by the vanishing of the discriminant $\Delta(z)$.
Pick a base point $z_1 \in \mathbf C-S$, and choose a root $w_1$ of $F(z_1, w)=0$. There are exactly $d$ roots to choose from. If $\sigma : I \to \mathbf C-S$ is a path starting at $z_0$, then there is a unique lift $\tilde \sigma$ of it to the $w$-plane, such that $\tilde \sigma(0) = w_1$ and $F(\sigma(t), \tilde\sigma(t))=0$ at each time $t$. What you must show is that any two points on $C-\tilde S$ can be connected by such a path $(\sigma(t), \tilde\sigma(t))$.
The problem is easily reduced to showing that any two points in the same fibre above a given $w$-point can be connected by such a path. I'll let you think about why this is possible. It is a good exercise (perhaps pick a simple $F$ where you can see what is going on). You'll have to choose paths which wind around the ramification points (the points of $S$) and come back to the base point, in order to travel from one point in the fibre to another.