Related papers: A Zariski dense exceptional set in Manin's conject…

We construct first examples of singular del Pezzo surfaces with Zariski dense exceptional sets in Manin's conjecture, varying in degrees $1, 2$ and $3$. The obstructions arise from accumulating quasi-\'etale covers. We classify all…

Batyrev and Tschinkel's example is a Fermat cubic surface bundle $X$ which is a Fano $5$-fold. It is the first example for which Manin's conjecture can never hold for a proper closed exceptional set. Recently, Lehmann, Sengupta, and…

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…

In this survey paper we study Manin's Conjecture from a geometric perspective. The focus of the paper is the recent conjectural description of the exceptional set in Manin's Conjecture due to Lehmann-Sengupta-Tanimoto. After giving an…

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 prove Manin's conjecture for a del Pezzo surface of degree six which has one singularity of type $\mathbf{A}_2$. Moreover, we achieve a meromorphic continuation and explicit expression of the associated height zeta function.

We prove that the integral points are potentially Zariski dense in the complement of a reduced effective singular anticanonical divisor in a smooth del Pezzo surface, with the exception of $\mathbb{P}^2$ minus three concurrent lines (for…

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 conjecture that the exceptional set in Manin's Conjecture has an explicit geometric description. Our proposal includes the rational point contributions from any generically finite map with larger geometric invariants. We prove that this…

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…

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

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 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 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…

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…

Let X be a surface whose Cox ring has a single relation satisfying moreover a kind of linearity property. Under a simple assumption, we show that the geometric Manin's conjectures hold for some degrees lying in the dual of the effective…

This is a report of the author's talk at Kinosaki Algebraic Geometry Symposium 2018. We discuss some recent progress on the geometry of thin exceptional sets in Manin's Conjecture.

Manin's Conjecture predicts the rate of growth of rational points of a bounded height after removing those lying on an exceptional set. We study whether the exceptional set in Manin's Conjecture is a thin set.

We prove Manin's conjecture for two del Pezzo surfaces of degree four which are split over Q and whose singularity types are respectively 3A_1 and A_1+A_2. For this, we study a certain restricted divisor function and use a result about the…

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