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The nef cone volume of generalized del Pezzo surfaces


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.

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
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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