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 |L| 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 the Göttsche-Yau-Zaslow Formula which expresses the generating function of the numbers of nodal curves in terms of quasi-modular forms and two unknown series.

arXiv comment29 pages

References in corpus (8)

• Algebraic cobordism
• A conjectural generating function for numbers of curves on surfaces
• Enumeration of $n$-fold tangent hyperplanes to a surface
• Severi Degrees in Cogenus 4
• The Algebraic Proof of the Universality Theorem
• Node polynomials for families: results and examples
• Algebraic cobordism of bundles on varieties
• Relative Node Polynomials for Plane Curves

Cited by in corpus (12)

• Reduced classes and curve counting on surfaces I: theory
• Universal Polynomials for Tautological Integrals on Hilbert Schemes
• Universal polynomials for singular curves on surfaces
• On the Göttsche Threshold
• A combinatorial analysis of Severi degrees
• Log BPS numbers of log Calabi-Yau surfaces
• Enumeration of singular varieties with tangency conditions
• The chi-y genera of relative Hilbert schemes for linear systems on Abelian and K3 surfaces
• Relative Node Polynomials for Plane Curves
• Algebraic cobordism of filtered vector bundles on varieties: Notes on a work of Lee and Pandharipande
• Gopakumar-Vafa BPS invariants, Hilbert schemes and quasimodular forms. I
• Curve Counting à la Göttsche