A proof of the Göttsche-Yau-Zaslow formula

Abstract

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