Enumeration of curves with one singular point

Abstract

In this paper we obtain an explicit formula for the number of curves in P 2 , of degree d , passing through ( d ( d + 3 ) / 2 − k ) generic points and having a singularity X , where X is of type A k ≤ 7 , D k ≤ 7 or E k ≤ 7 . Our method comprises of expressing the enumerative problem as the Euler class of an appropriate bundle and using a purely topological method to compute the degenerate contribution to the Euler class. These numbers have also been computed by M. Kazarian using the existence of universal formulas for Thom polynomials.