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

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.

Del Pezzo surface, Manin's conjecture, nef cone, root system
Mathematical Subject Classification 2000
Primary: 14J26
Secondary: 14C20, 14G05
Received: 27 July 2007
Revised: 19 October 2007
Accepted: 11 December 2007
Published: 15 March 2008
Ulrich Derenthal
Institut für Mathematik
Universität Zürich
Winterthurerstrasse 190
8057 Zürich
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