Related papers: Points of bounded height on quintic del Pezzo surf…

We prove Manin's conjecture for split smooth quintic del Pezzo surfaces over arbitrary number fields with respect to fairly general anticanonical height functions. After passing to universal torsors, we first show that we may restrict the…

An asymptotic formula is established for the number of rational points of bounded height on a non-singular quartic del Pezzo surface with a conic bundle structure.

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…

Manin's conjecture for the asymptotic behavior of the number of rational points of bounded height on del Pezzo surfaces can be approached through universal torsors. We prove several auxiliary results for the estimation of the number of…

Upper and lower bounds, of the expected order of magnitude, are obtained for the number of rational points of bounded height on any quartic del Pezzo surface over $\mathbb{Q}$ that contains a conic defined over $\mathbb{Q}$.

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

In this paper the height zeta function associated to a certain singular del Pezzo surface of degree four is studied. If $U$ denotes the open subset formed by deleting the unique line from this surface, then an asymptotic formula for the…

We discuss Manin's conjecture concerning the distribution of rational points of bounded height on Del Pezzo surfaces, and its refinement by Peyre, and explain applications of universal torsors to counting problems. To illustrate the method,…

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…

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.

We prove the "all-the-heights'' version of the Batyrev--Manin--Peyre conjecture for split quintic del Pezzo surfaces, both for counting rational points over global function fields in positive characteristic and for the motivic version over…

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 prove a lower bound that agrees with Manin's prediction for the number of rational points of bounded height on the Fermat cubic surface. As an application we provide a simple counterexample to Manin's conjecture over the rationals.

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…

We introduce the split torsor method to count rational points of bounded height on Fano varieties. As an application, we prove Manin's conjecture for all nonsplit quartic del Pezzo surfaces of type $\mathbf A_3+\mathbf A_1$ over arbitrary…

For split smooth Del Pezzo surfaces, we analyse the structure of the effective cone and prove a recursive formula for the value of alpha, appearing in the leading constant as predicted by Peyre of Manin's conjecture on the number of…

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…

We prove a version of Manin's conjecture (over $\mathbb{F}_{q}$ for $q$ large) and the Cohen--Jones--Segal conjecture (over $\mathbb{C}$) for maps from rational curves to split quartic del Pezzo surfaces. The proofs share a common method…

We prove that the number of rational points of bounded height on certain del Pezzo surfaces of degree 1 defined over Q grows linearly, as predicted by Manin's conjecture. Along the way, we investigate the average number of integral points…

We prove an asymptotic formula for the number of integral points of bounded log-anticanonical height on split smooth quintic del Pezzo surfaces over number fields, with respect to one of the lines as the boundary divisor.