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..."

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

This is how people used to think of Riemann surfaces. We've come a long way!


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