Rational points of bounded height on general 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 for 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.

Rational points and non-anticanonical height functions

“…We work over a field k, assumed to not have characteristic 2 for simplicity. The theory presented here is just a mild generalisation to higher dimensions of [FLS18,§2].…”

“…It proves, for the first time, a case of the Batyrev-Manin conjecture for smooth cubic surfaces with respect to a height function associated to some ample line bundle. (Facts about del Pezzo surfaces with a conic bundle structure can be found in [FLS18,§5]. )…”

Counting rational points on quartic del Pezzo surfaces with a rational conic

“…Invoking [15,Thm. 1.6], the lower bound in Theorem 1.1 is a direct consequence of the divisor sum conjecture that is recorded in [14, Con.…”

“…Leung [21] revisited Salberger's argument to promote the B ε to an explicit power of log B. On the other hand, recent work of Frei, Loughran and Sofos [15,Thm. 1.2] provides a lower bound for NpBq of the predicted order of magnitude for any quartic del Pezzo surface over Q with a Q-conic bundle structure and Picard rank ρ ě 4.…”

“…, G n has positive degree. Using [15], we shall see in §3 that our proof of the lower bound in Theorem 1.1 may proceed for surfaces X Ñ P 1 of Picard rank ρ " 2. In this case the fibre above any degenerate closed point of P 1 must be non-split by (1.1).…”