四度德尔皮佐表面上的有理点

Counting rational points on del Pezzo surfaces with a conic bundle structure

For any number field k, upper bounds are established for the number of k-rational points of bounded height on non-singular del Pezzo surfaces defined over k, which are equipped with suitable conic bundle structures over k.

An overview of Manin's conjecture for del Pezzo surfaces

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 singular del Pezzo surface of degree four.

Manin's conjecture for a quintic del Pezzo surface with A_2 singularity

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

Manin's conjecture for a quartic del Pezzo surface with A_4 singularity

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

del Pezzo曲面上的逆曲线问题

Inverse curve problems on del Pezzo surfaces

该研究根据相对极小曲面分类del Pezzo曲面上k-有理直线和二次曲线纤维化数量，建立其逆伽罗瓦问题的有理曲线类似问题，在各类域上完全解决所有次数的相关问题并推广了三次曲面定理。

Inverse Galois problem for del Pezzo surfaces over finite fields

We completely solve the inverse Galois problem for del Pezzo surfaces of degree $2$ and $3$ over all finite fields.

Del Pezzo surfaces of degree four violating the Hasse principle are Zariski dense in the moduli scheme

We show that, over every number field, the degree four del Pezzo surfaces that violate the Hasse principle are Zariski dense in the moduli scheme.

On Manin's conjecture for singular del Pezzo surfaces of degree four, II

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.

Singular del Pezzo surfaces that are equivariant compactifications

We determine which singular del Pezzo surfaces are equivariant compactifications of G_a^2, to assist with proofs of Manin's conjecture for such surfaces. Additionally, we give an example of a singular quartic del Pezzo surface that is an equivariant compactification of a semidirect product of G_a and G_m.

特征零域上四次 del Pezzo 曲面的自同构

Automorphisms of quartic del Pezzo surfaces in characteristic zero

本文对特征零域上四次 del Pezzo 曲面的自同构群进行完整分类，并区分有理、稳定有理与单有理情形，还为每个可实现群给出显式曲面方程。

On a constant arising in Manin's conjecture for Del Pezzo surfaces

For split smooth Del Pezzo surfaces, we analyse the structure of the effective cone and prove a recursive formula for the value of alpha, appearing in the leading constant as predicted by Peyre of Manin's conjecture on the number of rational points of bounded height on the surface. Furthermore, we calculate alpha for all singular Del Pezzo surfaces of degree at least 3.

Manin's conjecture for quartic del Pezzo surfaces with a conic fibration

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.

On the arithmetic of del Pezzo surfaces of degree 2

We study the arithmetic of certain del Pezzo surfaces of degree 2. We produce examples of Brauer-Manin obstruction to the Hasse principle, coming from 2- and 4-torsion elements in the Brauer group.

Manin's conjecture for a quartic del Pezzo surface with A_3 singularity and four lines

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 is not a hypersurface for which Manin's conjecture is proved.

具有Brauer-Manin障碍的2次对角del Pezzo曲面

Diagonal del Pezzo of degree 2 with a Brauer-Manin obstruction

该研究针对按高度排序的2次对角del Pezzo曲面，推导了存在Hasse原理的Brauer-Manin障碍的这类曲面数量的渐近公式。

Hilbert property for double conic bundles and del Pezzo varieties

In this paper we prove that, over a Hilbertian ground field, surfaces with two conic fibrations whose fibres have non-zero intersection product have the Hilbert property. We then give an application of this result, namely the verification of the Hilbert property for certain del Pezzo varieties.

Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields

We study the birational properties of geometrically rational surfaces from a derived categorical point of view. In particular, we give a criterion for the rationality of a del Pezzo surface over an arbitrary field, namely, that its derived category decomposes into zero-dimensional components. For del Pezzo surfaces of degree at least 5, we construct explicit semiorthogonal decompositions by subcategories of modules over semisimple algebras arising as endomorphism algebras of vector bundles and we show how to retrieve information about the index of the surface from Brauer classes and Chern clas…

关于度数为1的德尔皮佐表面的反伽罗瓦问题

On the inverse Galois problem for del Pezzo surfaces of degree 1

本文完全解决了度数为1的德尔皮佐表面在有限域上的反伽罗瓦问题，针对112种类型中的85种，并确定了所有类型的最小存在域。

On the number of certain Del Pezzo surfaces of degree four violating the Hasse principle

We give an asymptotic expansion for the density of del Pezzo surfaces of degree four in a certain Birch Swinnerton-Dyer family violating the Hasse principle due to a Brauer-Manin obstruction. Under the assumption of Schinzel's hypothesis and the finiteness of Tate-Shafarevich groups for elliptic curves, we obtain an asymptotic formula for the number of all del Pezzo surfaces in the family, which violate the Hasse principle.

Classification of singular del Pezzo surfaces over finite fields

In this article, we consider weak del Pezzo surfaces defined over a finite field, and their associated, singular, anticanonical models. We first define arithmetic types for such surfaces, by considering the Frobenius actions on their Picard groups; this extends the classification of Swinnerton-Dyer and Manin for ordinary del Pezzo surfaces. We also show that some invariants of the surfaces only depend on the above type.Then we study an inverse Galois problem for singular del Pezzo surfaces having degree $3\leq d\leq 6$: we describe which types can occur over a given finite field (of odd charac…

Rational points on certain del Pezzo surfaces of degree one

Let $f(z)=z^5+az^3+bz^2+cz+d \in \Z[z]$ and let us consider a del Pezzo surface of degree one given by the equation $\cal{E}_{f}: x^2-y^3-f(z)=0$. In this note we prove that if the set of rational points on the curve $E_{a, b}:Y^2=X^3+135(2a-15)X-1350(5a+2b-26)$ is infinite, then the set of rational points on the surface $\cal{E}_{f}$ is dense in the Zariski topology.

On the arithmetic of one del Pezzo surface over the field with three elements

We discuss the problem of existence of rational curves on a certain del Pezzo surface from a computational point of view and suggest a computer algorithm implementing search. In particular, our computations reveal that the surface contains 920 rational curves with parametrizations of degree 8 and does not contain rational curves for a smaller degree.

Distribution locale des points rationnels de hauteur bornée sur une surface de del Pezzo de degré 6

We establish a measure which describes in a precise way the local asymptotic distribution of rational points outside the locally accumulating subvarieties around a general rational point on a del Pezzo surface of degree 6 in the sense of diophantine approximation.

Bounds for del Pezzo surfaces of degree two

In this article, we obtain an upper bound for the number of integral points on the del Pezzo surfaces of degree two.

Manin's Conjecture for a Singular Sextic del Pezzo Surface

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.

具有圆锥纤维化结构的五次del Pezzo曲面的Manin猜想

Manin's conjecture for quintic del Pezzo surfaces with a conic bundle structure

本文研究具有圆锥纤维化结构的五次del Pezzo曲面的Manin猜想，通过证明匹配的上下界，并在Galois一般情形下完整验证了该猜想。