On the G\"ottsche Threshold
Algebraic Geometry
2013-02-08 v2
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.
Cite
@article{arxiv.1204.6254,
title = {On the G\"ottsche Threshold},
author = {Steven L. Kleiman and Vivek V. Shende and with an appendix by Ilya Tyomkin},
journal= {arXiv preprint arXiv:1204.6254},
year = {2013}
}
Comments
20 pages, final version, to appear in the Harris60 Volume of the Clay series (put out by the American Mathematical Society)