Density of rational points on diagonal bidegree $(1,2)$ hypersurface in $\mathbb{P}^{s-1} \times \mathbb{P}^{s-1}$

TL;DR

This paper proves an asymptotic formula for counting rational points of bounded height on a specific biprojective hypersurface, confirming the Manin conjecture for this class of varieties when dimension is at least 7.

Contribution

It establishes the Manin conjecture for a particular diagonal bidegree (1,2) hypersurface in biprojective space, providing a new case where the conjecture holds.

Findings

Derived an asymptotic formula for rational points count

Confirmed Manin conjecture for the hypersurface when s ≥ 7

Identified the density of rational points on the hypersurface

Abstract

In this paper we establish an asymptotic formula for the number of rational points of bounded anticanonical height which lie on a certain Zariski dense subset of the biprojective hypersurface \begin{align*} x_1y_1^2+...+x_sy_s^2 = 0 \end{align*} in $\mathbb{P}^{s-1} \times \mathbb{P}^{s-1} $ with $s \geq 7$. This confirms the Manin conjecture for this variety.