Rational points and non-anticanonical height functions

TL;DR

This paper advances the understanding of rational points on algebraic surfaces by proving new cases of the Batyrev-Manin conjecture for conic bundle surfaces and specific cubic surfaces with non-anticanonical heights.

Contribution

It establishes the conjecture for certain conic bundle surfaces and smooth cubic surfaces using non-anticanonical height functions, extending previous results.

Findings

Confirmed the Batyrev-Manin conjecture for specific conic bundle surfaces.

Verified the conjecture for some smooth cubic surfaces with particular ample line bundles.

Demonstrated the effectiveness of non-anticanonical height functions in counting rational points.

Abstract

A conjecture of Batyrev and Manin predicts the asymptotic behaviour of rational points of bounded height on smooth projective varieties over number fields. We prove some new cases of this conjecture for conic bundle surfaces equipped with some non-anticanonical height functions. As a special case, we verify these conjectures for the first time for some smooth cubic surfaces for height functions associated to certain ample line bundles.