Counting curves on a general linear system with up to two singular points

TL;DR

This paper derives an explicit formula for counting curves with specific singularities on complex surfaces, extending classical topological methods to cases with up to two singular points and codimension constraints.

Contribution

It provides a new explicit formula for enumerating curves with one node and one singularity of codimension up to 7 on complex surfaces.

Findings

Derived explicit formulas for curve counts with singularities

Extended classical topological methods to more complex singularity configurations

Applicable to enumerative geometry problems involving complex surfaces

Abstract

In this paper we obtain an explicit formula for the number of curves in a compact complex surface $X$ (passing through the right number of generic points), that has up to one node and one singularity of codimension $k$, provided the total codimension is at most $7$. We use a classical fact from differential topology: the number of zeros of a generic smooth section of a vector bundle $V$ over $M$, counted with signs, is the Euler class of $V$ evaluated on the fundamental class of $M$.