A Proof of the G\"ottsche-Yau-Zaslow Formula

TL;DR

This paper proves G"ottsche's conjecture on counting r-nodal curves on surfaces and establishes the G"ottsche-Yau-Zaslow formula linking these counts to modular forms, advancing algebraic geometry and enumerative theory.

Contribution

It provides a proof of G"ottsche's conjecture using algebraic cobordism and degeneration techniques, and derives the G"ottsche-Yau-Zaslow formula relating curve counts to modular forms.

Findings

Proof of G"ottsche's conjecture for r-nodal curves.

Derivation of the G"ottsche-Yau-Zaslow generating function.

Connection between curve counts and quasi-modular forms.

Abstract

Let S be a complex smooth projective surface and L be a line bundle on S. G\"ottsche conjectured that for every integer r, the number of r-nodal curves in |L| is a universal polynomial of four topological numbers when L is sufficiently ample. We prove G\"ottsche's conjecture using the algebraic cobordism group of line bundles on surfaces and degeneration of Hilbert schemes of points. In addition, we prove the the G\"ottsche-Yau-Zaslow Formula which expresses the generating function of the numbers of nodal curves in terms of quasi-modular forms and two unknown series.