2
$\begingroup$

Given a closed curve $C$ in the affine plane, we define its winding number around a point (that does not meet $C$) as the total number of times $C$ travels counter-clockwise around the point.

winding_-2 -2 winding_-1 -1 winding_0 0 winding_1 1 winding_2 2 winding_2 3

[images are from Wikipedia]

Suppose now that we are working over an algebraically closed field $k$ of characteristic zero. Do we have a similar notion for curves defined as the vanishing locus of a polynomial (that is, $C=V(f)$ with $f\in k[x,y]$ irreducible)?

Of course, it is not even clear to me how to tell whether $f$ defines a closed curve, and we have no orientation at all. Nevertheless, for regular curves, at least we have a plausible parametrisation, and for (the more interesting) singular ones, we could use blow-ups along the singular points.

As an example:

Consider the unit circle in $\mathbb{A}^2_{\mathbb{C}}$ given by $C=V(x^2+y^2-1)$. Then, for points inside the circle, the winding number should be one (we only care about the modulus), outside it should be zero.

  • 1
    The standard algebro-geometric way of dealing with the question of fundamental group is the etale fundamental group. Perhaps you can approach algebro-geometric winding numbers with a similar philosophy.2012-12-11

0 Answers 0