Universal polynomials for singular curves on surfaces

TL;DR

This paper proves that the count of algebraic curves with specified singularities on a smooth projective surface can be expressed as universal polynomials in the surface's Chern numbers, generalizing previous conjectures.

Contribution

It establishes the universality of polynomial formulas for counting singular curves with higher singularities, extending Gottsche's conjecture.

Findings

Number of singular curves is given by universal polynomials in Chern numbers.

Defined a generating series for these universal polynomials.

Generalized existing conjectures to include higher singularities.

Abstract

Let S be a complex smooth projective surface and L be a line bundle on S. For any given collection of isolated topological or analytic singularity types, we show the number of curves in the linear system |L| with prescribed singularities is a universal polynomial of Chern numbers of L and S, assuming L is sufficiently ample. Moreover, we define a generating series whose coefficients are these universal polynomials and discuss its properties. This work is a generalization of Gottsche's conjecture to curves with higher singularities.