Structure of Node Polynomials for Curves on Surfaces

Details Structure of Node Polynomials for Curves on Surfaces Structure of Node Polynomials for Curves on Surfaces

2011-02-10

arxiv.org/abs/1102.2092v2

Mathematics

Algebraic Geometry

20 pages. Overall revision, corrections of misprints, Section 6 shortened, some clarifications in the introduction

Scientific paper

We provide a non-numerical generalization of a theorem by Kleiman-Piene, concerning the enumerative geometry of nodal, algebraic curves in a complete linear system on a smooth projective surface S. Provided that the number of nodes is sufficiently small compared to the ampleness of the linear system, we show that the number of r-nodal curves passing through points in general position on S is given by a Bell polynomial in variables a_{i}, which we identify using classical intersection theory and express as linear, integral polynomials in four basic Chern numbers.

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