Universal polynomials for singular curves on surfaces

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.