Enumeration of curves with two singular points

TL;DR

This paper derives an explicit formula for counting degree d complex projective plane curves with specific singularities passing through a set number of points, using topological methods.

Contribution

It provides a new explicit enumeration formula for curves with two singular points, combining algebraic geometry with differential topology techniques.

Findings

Derived explicit formulas for enumerating curves with singularities.

Applied Euler class and differential topology to algebraic geometry problems.

Extended enumeration to curves with singularities up to codimension 6.

Abstract

In this paper we obtain an explicit formula for the number of curves in two dimensional complex projective space, 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.