Related papers: On Manin's conjecture for a family of Ch\^atelet s…

Following the line of attack from La Bret\`eche, Browning and Peyre, we prove Manin's conjecture in its strong form conjectured by Peyre for a family of Ch\^atelet surfaces which are defined as minimal proper smooth models of affine…

We resolve Manin's conjecture for all Ch\^atelet surfaces over $\mathbb{Q}$.

Generalising work of La Bret\`eche, Browning and Peyre and of the author, we describe the geometry required by Manin's principle and Peyre's conjecture for a family of conic bundle surfaces containing Ch\^atelet surfaces. These surfaces…

We establish Manin's conjecture for a cubic surface split over Q and whose singularity type is 2A_2+A_1. For this, we make use of a deep result about the equidistribution of the values of a certain restricted divisor function in three…

Manin's conjecture is proved for a split del Pezzo surface of degree 5 with a singularity of type A_2.

The Manin conjecture is established for a split singular cubic surface in P^3, with singularity type D_5.

The Manin-Peyre conjecture is established for a split singular quintic del Pezzo surface with singularity type $\mathbf{A}_2$ and two split singular quartic del Pezzo surfaces with singularity types $\mathbf{A}_3+\mathbf{A}_1$ and…

The Manin conjecture is established for a split singular del Pezzo surface of degree four, with singularity type A_4.

Given a nonsingular quartic del Pezzo surface, a conjecture of Manin predicts the density of rational points on the open subset of the surface formed by deleting the lines. We prove that this prediction is of the correct order of magnitude…

The Manin-Peyre conjecture is established for smooth spherical Fano threefolds of semisimple rank one and type N. Together with the previously solved case T and the toric cases, this covers all types of smooth spherical Fano threefolds. The…

We prove that any smooth cubic surface defined over any number field satisfies the lower bound predicted by Manin's conjecture possibly after an extension of small degree.

We establish Manin's conjecture for a quartic del Pezzo surface split over Q and having a singularity of type A_3 and containing exactly four lines. It is the first example of split singular quartic del Pezzo surface whose universal torsor…

Let S Q denote x 3 = Q(y 1 ,. .. , y m)z where Q is a primitive positive definite quadratic form in m variables with integer coefficients. This S Q ranges over a class of singular cubic hypersurfaces as Q varies. For S Q we prove (i)…

This paper establishes the Manin conjecture for a certain non-split singular del Pezzo surface of degree four, via an analysis of the corresponding height zeta function.

The Manin-Peyre conjecture is established for a class of smooth spherical Fano varieties of semisimple rank one. This includes all smooth spherical Fano threefolds of type T as well as some higher-dimensional smooth spherical Fano…

We investigate Manin's conjecture for del Pezzo surfaces of degree five with a conic bundle structure, proving matching upper and lower bounds, and the full conjecture in the Galois general case.

We give a relatively short and elementary proof of Manin's conjecture for split smooth quintic del Pezzo surfaces over the rational numbers.

This paper surveys recent progress towards the Manin conjecture for (singular and non-singular) del Pezzo surfaces. To illustrate some of the techniques available, an upper bound of the expected order of magnitude is established for a…

In this paper, we construct three kinds of Ch\^atelet surfaces, which have some given arithmetic properties with respect to field extensions of number fields. We then use these constructions to study the properties of weak approximation…

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…