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 with -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 -curves on . When is defined over an arbitrary perfect field, a similar result holds, except that the associated root system is no longer necessarily simply-laced.
"The nef cone volume of generalized Del Pezzo surfaces." Algebra Number Theory 2 (2) 157 - 182, 2008. https://doi.org/10.2140/ant.2008.2.157