Related papers: Manin's conjecture for a cubic surface with D_5 si…
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…
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.
The Manin-Peyre conjecture is established for smooth spherical Fano threefolds of semisimple rank one and type N. Together with the previously solved case T and the toric cases, this covers all types of smooth spherical Fano threefolds. 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.
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…
Using the circle method, we count integer points on complete intersections in biprojective space in boxes of different side length, provided the number of variables is large enough depending on the degree of the defining equations and…
We prove the thin set version of Manin's conjecture for the chordal (or: determinantal) cubic fourfold, which is the secant variety of the Veronese surface. We reduce this counting problem to a result of Schmidt for quadratic points in the…
We show that the number of non-trivial rational points of height at most $B$, that lie on the cubic surface $x_1x_2x_3=x_4(x_1+x_2+x_3)^2$, has order of magnitude $B(\log B)^6$. This agrees with the Manin conjecture.
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…
Using recent work of the first author~\cite{Bet}, we prove a strong version of the Manin-Peyre's conjectures with a full asymptotic and a power-saving error term for the two varieties respectively in $\mathbb{P}^2 \times \mathbb{P}^2$ with…
We prove the Tate conjecture for integral degree 4 classes on a smooth cubic hypersurface X of dimension 4 over an algebraic closure of a field finitely generated over its prime subfield.
We study singularities of surfaces which are given by Kenmotsu-type formula with prescribed unbounded mean curvature.
A strong form of the Manin-Peyre conjecture with a power saving error term is proved for a certain cubic fourfold.
We give a relatively short and elementary proof of Manin's conjecture for split smooth quintic del Pezzo surfaces over the rational numbers.
The Manin-Peyre conjecture is established for a class of smooth spherical Fano varieties of semisimple rank one. This includes all smooth spherical Fano threefolds of type T as well as some higher-dimensional smooth spherical Fano…
We resolve Manin's conjecture for all Ch\^atelet surfaces over $\mathbb{Q}$.
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 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.
In this note, we establish an asymptotic formula for the number of rational points of bounded height on the singular cubic surface $$ x_0(x_1^2 + x_2^2)=x_3^3 $$ with a power-saving error term, which verifies the Manin-Peyre conjectures for…
We prove a version of Manin's conjecture for a certain family of intrinsic quadrics, the base field being a global field of positive characteristic. We also explain how a very slight variation of the method we use allows to establish the…