Enumeration of curves with one singular point

TL;DR

This paper derives explicit formulas for counting degree d curves with a single singular point in complex projective space, using topological methods instead of traditional algebraic geometry techniques.

Contribution

It introduces a purely topological approach to enumerate curves with singularities, providing explicit formulas for cases with up to seven singularity codimension.

Findings

Derived explicit formulas for curve counts with singularities

Applied topological methods to algebraic geometry problems

Extended results to curves with singularities of codimension up to 7

Abstract

In this paper we obtain an explicit formula for the number of degree d curves in two dimensional complex projective space, passing through (d(d+3)/2 -k) generic points and having a codimension k singularity, where k is at most 7. In the past, many of these numbers were computed using techniques from algebraic geometry. In this paper we use purely topological methods to count curves. 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.