Rational points and non-anticanonical height functions

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.Comment: 16 pages; minor corrections; Proceedings of the American Mathematical Society, 147 (2019), no. 8, 3209-322

On a certain non-split cubic surface

In this note, we establish an asymptotic formula for the number of rational points of bounded height on the singular cubic surface $$x_0(x_1^2 + x_2^2)=x_3^3$$ with a power-saving error term, which verifies the Manin-Peyre conjectures for this surface.Comment: v2. 18 pages, one author adde

On the order three Brauer classes for cubic surfaces

We describe a method to compute the Brauer-Manin obstruction for smooth cubic surfaces over \bbQ such that \Br(S)/\Br(\bbQ) is a 3-group. Our approach is to associate a Brauer class with every ordered triplet of Galois invariant pairs of Steiner trihedra. We show that all order three Brauer classes may be obtained in this way. To show the effect of the obstruction, we give explicit examples.Comment: Extended introduction, discussion of the example of Cassels and Gu

The Manin conjecture in dimension 2

These lecture notes describe the current state of affairs for Manin's conjecture in the context of del Pezzo surfaces.Comment: 57 pages. These are a preliminary version of lecture notes for the "School and conference on analytic number theory", ICTP, Trieste, 23/04/07-11/05/0

Rational points of bounded height on general conic bundle surfaces

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.Comment: 35 pages; final versio