Integral points over number fields: a Clemens complex jigsaw puzzle

TL;DR

This paper establishes an asymptotic count of integral points on a singular quartic del Pezzo surface over number fields, revealing a complex Clemens complex structure and novel effective cone constants through a universal torsor approach.

Contribution

It introduces a detailed analysis of the Clemens complex for singular surfaces and derives an asymptotic formula involving complex polytope volumes using o-minimal structures.

Findings

Asymptotic formula for integral points on the surface.

Identification of complex polytope volume contributions.

Novel application of o-minimal structures in counting problems.

Abstract

We prove an asymptotic formula for the number of integral points of bounded log anticanonical height on a singular quartic del Pezzo surface over arbitrary number fields, with respect to the largest admissible boundary divisor. The resulting Clemens complex is more complicated than usual, and leads to particularly interesting effective cone constants, associated with exponentially many polytopes whose volumes appear in the expected formula. Like a jigsaw puzzle, these polytopes fit together to one large polytope. The volume of this polytope appears in the asymptotic formula that we obtain using the universal torsor method via o-minimal structures.