Density of rational points on some quadric bundle threefolds

TL;DR

This paper proves the Manin-Peyre conjecture for rational points on certain Fano threefolds, combining the circle method and geometry of numbers to establish precise asymptotics.

Contribution

It introduces a novel approach by integrating the circle method with geometry of numbers to verify the Manin-Peyre conjecture for specific quadric bundle threefolds.

Findings

Confirmed the Manin-Peyre conjecture for the studied threefolds.

Established asymptotic formulas for the count of rational points.

Demonstrated the effectiveness of combined analytic and geometric techniques.

Abstract

We prove the Manin-Peyre conjecture for the number of rational points of bounded height outside of a thin subset on a family of Fano threefolds of bidegree (1,2). The proof uses a mixture of the circle method and techniques from the geometry of numbers.