Complex algebraic plane curves via the Poincaré–Hopf formula, I: Parametric lines

Maciej Borodzik and Henryk Żołądek

Vol. 229 (2007), No. 2, 307–338

DOI: 10.2140/pjm.2007.229.307

Abstract

We present a new and very efficient approach to study topology of algebraic curves in ℂ². It relies on using the Poincaré–Hopf formula, applied to a suitable Hamiltonian vector field, to estimate the number of double points and the Milnor numbers of singular points of the curve and on considering finite-dimensional spaces of curves with given asymptotics at infinity. We apply this method to classification of parametric lines with one self-intersection.

Keywords

affine algebraic curve, index of a vector field, Puiseux expansion

Mathematical Subject Classification 2000

Primary: 14H50

Secondary: 32S05, 14H15, 14H20

Milestones

Received: 18 July 2005

Accepted: 7 December 2005

Published: 1 February 2007