On the G\"ottsche Threshold

TL;DR

This paper extends G"ottsche's conjecture on the degree of Severi varieties to broader classes of rational surfaces, showing that certain codimension conditions suffice for the polynomial formula to hold.

Contribution

It relaxes the ample condition on line bundles for rational surfaces, proving the conjecture for P^2, Hirzebruch, and del Pezzo surfaces under new codimension criteria.

Findings

G"ottsche's conjecture holds for P^2 and Hirzebruch surfaces.

A similar condition applies to classical del Pezzo surfaces.

Degree of Severi varieties is given by a universal polynomial under these conditions.

Abstract

For a line bundle L on a smooth surface S, it is now known that the degree of the Severi variety of cogenus-d curves is given by a universal polynomial in the Chern classes of L and S if L is d-very ample. For S rational, we relax the latter condition substantially: it suffices that three key loci be of codimension more than d. As corollaries, we prove that the condition conjectured by G\"ottsche suffices if S is P^2 or S is any Hirzebruch surface, and that a similar condition suffices if S is any classical del Pezzo surface.