The nef cone volume of generalized Del Pezzo surfaces
Abstract
We compute a naturally defined measure of the size of the nef cone of a Del Pezzo surface. The resulting number appears in a conjecture of Manin on the asymptotic behavior of the number of rational points of bounded height on the surface. The nef cone volume of a Del Pezzo surface Y with (-2)-curves defined over an algebraically closed field is equal to the nef cone volume of a smooth Del Pezzo surface of the same degree divided by the order of the Weyl group of a simply-laced root system associated to the configuration of (-2)-curves on Y. When Y is defined over a non-closed field of characteristic 0, a similar result holds, except that the associated root system is no longer necessarily simply-laced.
Keywords
Cite
@article{arxiv.math/0703202,
title = {The nef cone volume of generalized Del Pezzo surfaces},
author = {Ulrich Derenthal and Michael Joyce and Zach Teitler},
journal= {arXiv preprint arXiv:math/0703202},
year = {2009}
}
Comments
v2, 25 pages; major revision containing additional results; comments welcome