On the factor alpha in Peyre's constant

TL;DR

This paper computes the constant alpha(S) for del Pezzo surfaces, which is crucial for understanding the distribution of rational points, using computational tools like Magma and Polymake.

Contribution

It provides a method to explicitly calculate alpha(S) for any del Pezzo surface, advancing the verification of Peyre's conjecture.

Findings

Computed alpha(S) for various del Pezzo surfaces

Validated Peyre's conjecture for these surfaces

Developed computational approach using Magma and Polymake

Abstract

For an arbitrary del Pezzo surface S, we compute alpha(S), which is the volume of a certain polytope in the dual of the effective cone of S, using Magma and Polymake. The constant alpha(S) appears in Peyre's conjecture for the leading term in the asymptotic formula for the number of rational points of bounded height on S over number fields.