Structure of Node Polynomials for Curves on Surfaces

TL;DR

This paper generalizes a theorem on counting nodal curves on surfaces, showing that for small numbers of nodes, these counts are given by Bell polynomials in universal integers related to Chern numbers.

Contribution

It introduces a structural framework for node polynomials, expressing them as Bell polynomials in universal integers and relating them to Thom polynomials for multisingularities.

Findings

Number of r-nodal curves is given by Bell polynomials in universal integers.

Universal integers are linear, integral polynomials in four Chern numbers.

Decomposition of these polynomials reveals geometric interpretations.

Abstract

We provide a structural generalization of a theorem by Kleiman--Piene, concerning the enumerative geometry of nodal curves in a complete linear system |L| on a smooth projective surface S. Provided that r, the number of nodes, is sufficiently small compared to the ampleness of the linear system, we show that, under certain assumptions, the number of r-nodal curves passing through points in general position on S is given by a Bell polynomial in universally defined integers a_i(S,L), which we identify, using classical intersection theory, as linear, integral polynomials evaluated in four basic Chern numbers. Furthermore, we provide a decomposition of the a_i as a sum of three terms with distinct geometric interpretations, and discuss the relationship between these polynomials and Kazarian's Thom polynomials for multisingularities of maps.