Node Polynomials for Curves on Surfaces

TL;DR

This paper completes the proof of a theorem on enumerating nodal curves on surfaces, showing that for up to 8 nodes, the cycle class can be expressed as a universal polynomial in Chern classes.

Contribution

It finalizes the proof of a previously announced theorem, establishing that for up to 8 nodes, the cycle class is given by a computable universal polynomial.

Findings

The cycle class for curves with up to 8 nodes is given by a universal polynomial.

The proof completes the enumeration of nodal curves on surfaces.

The result provides a computable formula for the cycle class in terms of Chern classes.

Abstract

We complete the proof of a theorem we announced and partly proved in [Math. Nachr. 271 (2004), 69-90, math.AG/0111299]. The theorem concerns a family of curves on a family of surfaces. It has two parts. The first was proved in that paper. It describes a natural cycle that enumerates the curves in the family with precisely $r$ ordinary nodes. The second part is proved here. It asserts that, for $r\le 8$, the class of this cycle is given by a computable universal polynomial in the pushdowns to the parameter space of products of the Chern classes of the family.