The nef cone volume of generalized Del Pezzo surfaces

Derenthal, Ulrich; Joyce, Michael; Teitler, Zachariah

doi:10.2140/ant.2008.2.157

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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

DOI: 10.2140/ant.2008.2.157

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.

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

Vol. 2, No. 2, 2008