Related papers: Manin's conjecture for toric varieties

We prove Manin's conjecture over imaginary quadratic number fields for a cubic surface with a singularity of type E_6.

We prove asymptotic formulas for the number of rational points of bounded height on certain blow-ups of the projective space.

Vojta's Conjectures are well known to imply a wide range of results, known and unknown, in arithmetic geometry. In this paper, we add to the list by proving that they imply that rational points tend to repel each other on algebraic…

We estimate the growth rate of the function which counts the number of torsion points of order at most $T$ on an algebraic subvariety of the algebraic torus $\mathbb G_m^n$ over some algebraically closed field. We prove a general upper…

We study rational points on conic bundles over elliptic curves with positive rank over a number field. We show that the etale Brauer-Manin obstruction is insufficient to explain failures of the Hasse principle for such varieties. We then…

Inspired by a method of La Bret\`eche relying on some unique factorisation, we generalize work of Blomer, Br\"udern, and Salberger to prove Manin's conjecture in its strong form conjectured by Peyre for some infinite family of varieties of…

We prove the Parshin's conjecture on the rational triviality of the higher algebraic $K$-theory of smooth projective varieties over finite fields. This is known to imply the Beilinson-Soul\'e conjecture for the fields of positive…

We give a short proof of the Zariski-Lipman conjecture for toric varieties: any complex toric variety with locally free tangent sheaf is smooth.

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…

Let V be a plane smooth cubic curve over a finitely generated field k. The Mordell-Weil theorem for V states that there is a finite subset P \subset V(k) such that the whole V(k) can be obtained from P by drawing secants and tangents…

Let $C$ be an algebraic curve embedded transversally in a power $E^N$ of an elliptic curve $E$. In this article we produce a good explicit bound for the height of all the algebraic points on $C$ contained in the union of all proper…

This paper initiates the systematic study of the number of points of bounded height on symmetric squares of weak Fano varieties. We provide a general framework for establishing the point count on $\text{Sym}^2 X$. In the specific case of…

We propose an empirical formula for the problem of local distribution of rational points of bounded height. This is a local version of the Batyrev-Manin-Peyre principle. We verify this for a toric surface, on which cuspidal rational curves…

Let C be a curve of genus at least 2 imbedded in a product of elliptic curves. We give an explicit upper bound for the points in the intersection of C with the union of all algebraic subgroups of a certain codimension. As a corollary we…

We give an upper bound on the number of rational points of an arbitrary Zariski closed subset of a projective space over a finite field. This bound depends only on the dimensions and degrees of the irreducible components and holds for very…

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…

We present an explicit formula for the canonical height of a projective toric variety.

In this paper, we complete the long-standing challenge to establish a Khintchine-type theorem for arbitrary nondegenerate manifolds in $\mathbb{R}^n$. In particular, our main result finally removes the analyticity assumption from the…

We solve a motivic version of the Adams conjecture with the exponential characteristic of the base field inverted. In the way of the proof we obtain a motivic version of mod k Dold theorem and give a motivic version of Brown's trick…

In this article, we establish the arithmetic purity of strong approximation for smooth loci of weighted projective spaces. By using this result and the descent method, we also prove that the arithmetic purity of strong approximation with…