Counting imaginary quadratic points via universal torsors

TL;DR

This paper develops a framework using universal torsors to study the distribution of rational points on del Pezzo surfaces over imaginary quadratic fields, advancing the understanding of Manin's conjecture in this context.

Contribution

It introduces a new method for proving Manin's conjecture for del Pezzo surfaces over imaginary quadratic fields, applicable to a broad class of number fields.

Findings

Proves Manin's conjecture for a specific quartic del Pezzo surface over imaginary quadratic fields.

Develops tools over arbitrary number fields for counting rational points.

Provides a general framework for similar proofs on other Fano varieties.

Abstract

A conjecture of Manin predicts the distribution of rational points on Fano varieties. We provide a framework for proofs of Manin's conjecture for del Pezzo surfaces over imaginary quadratic fields, using universal torsors. Some of our tools are formulated over arbitrary number fields. As an application, we prove Manin's conjecture over imaginary quadratic fields K for the quartic del Pezzo surface S of singularity type A_3 with five lines given in P^4 by the equations vw - xy = vy + wy + xz = 0.