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

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.