Integral Points on a del Pezzo Surface over Imaginary Quadratic Fields

TL;DR

This paper characterizes and counts integral points of bounded height on a specific quartic del Pezzo surface over imaginary quadratic fields, extending Manin's conjecture to integral points with geometric interpretations.

Contribution

It provides a detailed characterization and counting method for integral points on a singular del Pezzo surface over imaginary quadratic fields, using universal torsors.

Findings

Established a count of integral points of bounded height

Proved an analogue of Manin's conjecture for these points

Provided geometric interpretation of the counting results

Abstract

We characterise integral points of bounded log-anticanonical height on a quartic del Pezzo surface of singularity type $\mathbf{A}_3$ over imaginary quadratic fields with respect to its singularity and its lines. Furthermore, we count these integral points of bounded height by using universal torsors and interpret the count geometrically to prove an analogue of Manin's conjecture for the set of integral points with respect to the singularity and to a line.