Enumeration of curves with two singular points

Abstract

In this paper we obtain an explicit formula for the number of curves in P2, of degree d, passing through (d(d+3)/2−(k+1)) generic points and having one node and one codimension k singularity, where k is at most 6. Our main tool is a classical fact from differential topology: the number of zeros of a generic smooth section of a vector bundle V over M, counted with a sign, is the Euler class of V evaluated on the fundamental class of M.