The nef cone volume of generalized Del Pezzo surfaces

Details The nef cone volume of generalized Del Pezzo surfaces The nef cone volume of generalized Del Pezzo surfaces

2007-03-07

arxiv.org/abs/math/0703202v2

Algebra & Number Theory 2 (2008), no. 2, 157--182

Mathematics

Algebraic Geometry

v2, 25 pages; major revision containing additional results; comments welcome

Scientific paper

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 a non-closed field of characteristic 0, a similar result holds, except that the associated root system is no longer necessarily simply-laced.

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