## Algebra & Number Theory

### 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 $(−2)$-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.

#### Article information

Source
Algebra Number Theory, Volume 2, Number 2 (2008), 157-182.

Dates
Revised: 19 October 2007
Accepted: 11 December 2007
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513797228

Digital Object Identifier
doi:10.2140/ant.2008.2.157

Mathematical Reviews number (MathSciNet)
MR2377367

Zentralblatt MATH identifier
1158.14032

#### Citation

Derenthal, Ulrich; Joyce, Michael; Teitler, Zachariah. The nef cone volume of generalized Del Pezzo surfaces. Algebra Number Theory 2 (2008), no. 2, 157--182. doi:10.2140/ant.2008.2.157. https://projecteuclid.org/euclid.ant/1513797228

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