Density of rational points on a quadric bundle in $\mathbb{P}^3\times \mathbb{P}^3$

TL;DR

This paper establishes an asymptotic formula for the density of rational points on a specific biprojective hypersurface, confirming a modified version of the Manin conjecture that accounts for thin sets.

Contribution

It provides the first asymptotic count of rational points on a particular quadric bundle in biprojective space, verifying the modified Manin conjecture for this case.

Findings

Asymptotic formula for rational points established

Confirms the modified Manin conjecture for the hypersurface

Removes a thin set of rational points in the count

Abstract

An asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariski dense subset of the biprojective hypersurface $x_1y_1^2+\dots+x_4y_4^2=0$ in $\mathbb{P}^3\times\mathbb{P}^3$. This confirms the modified Manin conjecture for this variety, in which the removal of a thin set of rational points is allowed.