#### Vol. 2, No. 2, 2008

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The nef cone volume of generalized Del Pezzo surfaces

### Ulrich Derenthal, Michael Joyce and Zachariah Teitler

Vol. 2 (2008), No. 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 $\left(-2\right)$-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 $\left(-2\right)$-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.

##### Keywords
Del Pezzo surface, Manin's conjecture, nef cone, root system
##### Mathematical Subject Classification 2000
Primary: 14J26
Secondary: 14C20, 14G05
##### Milestones
Received: 27 July 2007
Revised: 19 October 2007
Accepted: 11 December 2007
Published: 15 March 2008
##### Authors
 Ulrich Derenthal Institut für Mathematik Universität Zürich Winterthurerstrasse 190 8057 Zürich Switzerland Michael Joyce Department of Mathematics Tulane University Gibson Hall 424 New Orleans, LA 70118 United States Zachariah Teitler Department of Mathematics Southeastern Louisiana University SLU 10687 Hammond, LA 70402 United States