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 an arbitrary perfect field, a similar result holds, except that the associated root system is no longer necessarily simply-laced.

Derenthal, U; Michael, J; Teitler, Z (2008). *The nef cone volume of generalized del Pezzo surfaces.* Algebra & Number Theory, 2(2):157-182.

## 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 an arbitrary perfect field, a similar result holds, except that the associated root system is no longer necessarily simply-laced.

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## Additional indexing

Item Type: | Journal Article, refereed, original work |
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Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Language: | English |

Date: | 2008 |

Deposited On: | 26 Feb 2009 06:47 |

Last Modified: | 05 Apr 2016 13:04 |

Publisher: | Mathematical Sciences Publishers |

ISSN: | 1937-0652 |

Publisher DOI: | https://doi.org/10.2140/ant.2008.2.157 |

Related URLs: | http://www.ams.org/mathscinet-getitem?mr=2377367 http://arxiv.org/abs/math/0703202v2 |

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