Rational points on singular intersections of quadrics

TL;DR

This paper investigates the distribution of rational points on the intersection of two quadrics in projective space, focusing on cases with conjugate singular points over the Gaussian field, especially in high-dimensional settings.

Contribution

It provides new insights into the behavior of rational points on singular intersections of quadrics, particularly when singular points are conjugate over Gaussian fields.

Findings

Quantitative analysis of rational points on singular intersections

Conditions under which rational points are dense or sparse

Impact of conjugate singular points on rational point distribution

Abstract

Given a projective intersection of two quadrics X in at least 9 variables, the quantitative behaviour of the rational points on X is investigated under the assumption that X contains a pair of conjugate singular points defined over the Gaussians.