Related papers: Motivic counting of curves on split quintic del Pe…
We give a relatively short and elementary proof of Manin's conjecture for split smooth quintic del Pezzo surfaces over the rational numbers.
Using the homological sieve method developed by Das--Lehmann--Tosteson and the author, we prove Peyre's all height approach to Manin's conjecture for split quintic del Pezzo surfaces defined over $\mathbb F_q(t)$ assuming $q$ is…
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 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…
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…
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…
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.
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 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…
Let $X$ be a del Pezzo surface over the function field of a complex curve. We study the behavior of rational points on $X$ leading to bounds on the counting function in Geometric Manin's Conjecture. A key tool is the Movable Bend and Break…
We give upper bounds for the number of rational points of bounded anti-canonical height on del Pezzo surfaces of degree at most five over any global field whose characteristic is not equal to two or three. For number fields these results…
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 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 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 a motivic version of the Poisson formula on the adelic points of a split algebraic torus and apply it to the study of the motivic height zeta function of split projective toric varieties, in the context of the motivic Manin-Peyre…
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…
We study the space of rational curves on del Pezzo surfaces in positive characteristic. For most primes p we prove the irreducibility of the moduli space of rational curves of a given nef class, extending results of Testa in characteristic…
Using the formalism of Cox rings and universal torsors, we prove a decomposition of the Grothendieck motive of the moduli space of morphisms from an arbitrary smooth projective curve to a Mori Dream Space (MDS). For the simplest cases of…
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…
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…