Rational points of bounded height on general conic bundle surfaces

TL;DR

This paper advances the understanding of rational points on conic bundle surfaces, providing new lower bounds that support Manin's conjecture for broad classes of del Pezzo surfaces over number fields.

Contribution

It introduces methods using conic bundles to establish lower bounds for rational points, extending the cases where Manin's conjecture is verified.

Findings

Confirmed Manin's conjecture lower bounds for del Pezzo surfaces with high Picard rank

Extended results to arbitrary del Pezzo surfaces over small degree extensions

Provided new techniques for analyzing rational points on conic bundle surfaces

Abstract

A conjecture of Manin predicts the asymptotic distribution of rational points of bounded height on Fano varieties. In this paper we use conic bundles to obtain correct lower bounds or a wide class of surfaces over number fields for which the conjecture is still far from being proved. For example, we obtain the conjectured lower bound of Manin's conjecture for any del Pezzo surface whose Picard rank is sufficiently large, or for arbitrary del Pezzo surfaces after possibly an extension of the ground field of small degree.