Related papers: Counting imaginary quadratic points via universal …
A conjecture of Manin predicts the distribution of rational points on Fano varieties. We provide a framework for proofs of Manin's conjecture for del Pezzo surfaces over imaginary quadratic fields, using universal torsors. Some of our tools…
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…
Inspired by a paper of Salberger we give a new proof of Manin's conjecture for toric varieties over imaginary quadratic number fields by means of universal torsor parameterizations and elementary lattice point counting.
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,…
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…
We characterise integral points of bounded log-anticanonical height on a quartic del Pezzo surface of singularity type $\mathbf{A}_3$ over imaginary quadratic fields with respect to its singularity and its lines. Furthermore, we count these…
Manin's conjecture predicts the distribution of rational points on Fano varieties. Using explicit parameterizations of rational points by integral points on universal torsors and lattice-point-counting techniques, it was proved for several…
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 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…
We prove Manin's conjecture for a split singular quartic del Pezzo surface with singularity type $2\Aone$ and eight lines. This is achieved by equipping the surface with a conic bundle structure. To handle the sum over the family of conics,…
We prove Manin's conjecture over imaginary quadratic number fields for a cubic surface with a singularity of type E_6.
A conjecture of Manin predicts the distribution of K-rational points on certain algebraic varieties defined over a number field K. In recent years, a method using universal torsors has been successfully applied to several hard special cases…
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…
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…
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.
Manin's conjecture predicts the asymptotic behavior of the number of rational points of bounded height on algebraic varieties. For toric varieties, it was proved by Batyrev and Tschinkel via height zeta functions and an application of the…
We classify all generalized del Pezzo surfaces (i.e., minimal desingularizations of singular del Pezzo surfaces containing only rational double points) whose universal torsors are open subsets of hypersurfaces in affine space. Equivalently,…
We give a relatively short and elementary proof of Manin's conjecture for split smooth quintic del Pezzo surfaces over the rational numbers.
In his book "Cubic forms" Manin discovered that del Pezzo surfaces are related to root systems. To explain the many numerical coincidences Batyrev conjectured that a universal torsor on a del Pezzo surface can be embedded in a certain…
The Manin conjecture is established for a split singular del Pezzo surface of degree four, with singularity type A_4.