The nef cone volume of generalized Del Pezzo surfaces

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.