T. D. Browning
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.
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Manin's conjecture is proved for a split del Pezzo surface of degree 5 with a singularity of type A_2.
The Manin conjecture is established for a split singular del Pezzo surface of degree four, with singularity type A_4.
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 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.
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。
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.
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.
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 number of rational points of bounded height on $U$ is established which verifies the Manin conjecture.
We show that, for every integer $1 \leq d \leq 4$ and every finite set $S$ of places, there exists a degree $d$ del Pezzo surface $X$ over ${\mathbb Q}$ such that ${\rm Br}(X)/{\rm Br}({\mathbb Q}) \cong {\mathbb Z}/2{\mathbb Z}$ and the Brauer-Manin obstruction works exactly at the places in $S$. For $d = 4$, we prove that in all cases, with the exception of $S = \{\infty\}$, this surface may be chosen diagonalizably over ${\mathbb Q}$.
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 equidistribution of its values in arithmetic progressions. In this task, Weil's bound for Kloosterman sums plays a key role.
The Manin-Peyre conjecture for three del Pezzo surfaces
本文证明了三个奇异 del Pezzo 曲面的 Manin-Peyre 猜想，采用统一的新方法改进了误差项，并指出该方法可推广至大多数次数不低于 3 的奇异 del Pezzo 曲面。
On Manin's conjecture for quartic del Pezzo fibrations
该研究推广同伦筛法，结合三维正特征极小模型纲领等工具，证明整体函数域上特定四次 del Pezzo 纤维化的 Manin 猜想，明确其几何性质。
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.
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。
Quasi-étale covers of Du Val del Pezzo surfaces and Zariski dense exceptional sets in Manin's conjecture
本文构造了 Manin 猜想中具有 Zariski 稠密例外集的奇异 del Pezzo 曲面首批例子（次数为1、2和3），通过分类 Du Val del Pezzo 曲面的拟平展覆盖并研究群作用排除了高次情况，提供了系统构造方法。
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 a result of independent interest on a certain restricted divisor problem for four binary linear forms.
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 provide a proof of Manin's conjecture for the unique split singular quartic Del Pezzo surface with a singularity of type D4. (Lectures at the summer school "Equidistribution in number theory", Montreal, August 2005)
Points of bounded height on quintic del Pezzo surfaces over the rational numbers
本文针对有理数域上分裂光滑五次del Pezzo曲面，给出了Manin猜想的一个相对简短且初等的证明。
Betti bounds for spaces of curves on varieties and Manin's conjecture for quartic del Pezzo surfaces
该研究证明了射影簇上固定亏格曲线态射空间的紧支撑贝蒂数的一致指数界，并将其应用于建立分裂四次德尔佩佐曲面马宁猜想的高亏格函数域版本。
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.
These lecture notes describe the current state of affairs for Manin's conjecture in the context of del Pezzo surfaces.
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 integral points in certain regions on universal torsors. As an application, we prove Manin's conjecture for a singular quartic del Pezzo surface.
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 are formulated over arbitrary number fields. As an application, we prove Manin's conjecture over imaginary quadratic fields K for the quartic del Pezzo surface S of singularity type A_3 with five lines given in P^4 by the equations vw - xy = vy + wy + xz = 0.
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 conjecture for a certain generalized del Pezzo surface.
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 specific varieties over $\mathbb{Q}$, in particular del Pezzo surfaces. We show how this method can be implemented over arbitrary number fields $K$, by proving Manin's conjecture for a singular quartic del Pezzo surface of type $\mathbf{A}_3+\mathbf{A}_1$. The parameterization step is treated in high generality with the help of twisted integral models of uni…
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.
Rational points on del Pezzo surfaces of degree four
本文研究四度德尔皮佐表面的有理点分布及布雷厄姆-曼因障碍对哈塞原理和弱逼近的影响。
We study arithmetic properties of del Pezzo surfaces of degree 4 for which the Brauer group has the largest possible order using different fibrations into curves. We show that if such a surface admits a conic fibration, then it always has a rational point. We also answer a question of Várilly-Alvarado and Viray by showing that the Brauer groups these surfaces cannot be vertical with respect to any projection away from a plane. We conclude that the available techniques for proving existence of rational points or even Zariski density do not directly apply if there is no Brauer-Manin obstruction …
Motivic counting of curves on split quintic del Pezzo surfaces
本文证明了分裂五次德尔皮佐表面的Batyrev-Manin-Peyre猜想的'所有高度'版本，涵盖了正特征全局函数域上的有理点计数及一般基域上的动机版本。
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 number fields. The counting problem on the split torsor is solved in the framework of o-minimal structures.
Homological stability and Manin's conjecture for rational curves on quartic del Pezzo surfaces
本文证明了有理曲线到分裂四次 del Pezzo 曲面的 Manin 猜想和 Cohen--Jones--Segal 猜想，通过 bar 复形、维数估计和虚拟高度 zeta 函数等方法，验证了 Batyrev 等人的启发式方法。
For given non-zero integers a,b,q we investigate the density of integer solutions (x,y) to the binary cubic congruence ax^2+by^3=0 (mod q). We use this to establish the Manin conjecture for a singular del Pezzo surface of degree 2 defined over the rationals and to examine the distribution of elliptic curves with square-free discriminant.
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.
Galois group of exceptional curves on the generic del Pezzo surface
研究了通用del Pezzo表面度数d的例外曲线的Galois作用，证明其为最大，并推导出在F_q(u)上立方面无Brauer-Manin障碍的结论。
Points of bounded height on quintic del Pezzo surfaces over number fields
该研究证明了任意数域上分裂光滑五次 del Pezzo 曲面关于一般反典范高函数的 Manin 猜想，通过泛挠子限制变量至典型大小并在 o-极小结构框架下解决计数问题。
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…
A conjecture of Manin predicts the asymptotic distribution of rational points of bounded height on Fano varieties. In this paper we use conic bundles to obtain correct lower bounds or a wide class of surfaces over number fields for which the conjecture is still far from being proved. For example, we obtain the conjectured lower bound of Manin's conjecture for any del Pezzo surface whose Picard rank is sufficiently large, or for arbitrary del Pezzo surfaces after possibly an extension of the ground field of small degree.
We report on our investigations concerning algebraic and transcendental Brauer-Manin obstructions to integral points on complements of a hyperplane section in degree four del Pezzo surfaces. We discuss moreover two concepts of an obstruction at an archimedean place. Concrete examples are given of pairs of non-homogeneous quadratic polynomials in four variables representing (0,0) over Q and over Z_p for all primes p, but not over Z. By blow-up, these yield cubic polynomials in three variables all integral solutions of which satisfy a gcd condition.
Quartic del Pezzo surfaces over $\mathbb{F}_p(t)$ without quadratic points
研究构造了无二次点的四次德尔皮佐表面，并展示了Brauer--Manin障碍。
We study del Pezzo surfaces of degree 1 of the form w^2 = z^3 + Ax^6 + By^6 in the weighted projective space P_k(1,1,2,3), where k is a perfect field of characteristic not 2 or 3 and A,B \in k^*. Over a number field, we exhibit an infinite family of (minimal) counterexamples to weak approximation amongst these surfaces, via a Brauer-Manin obstruction.
The spaces of rational curves on del Pezzo surfaces via conic bundles
本文利用Das-Lehmann-Tosteson等人发展出的同调筛方法，证明了分裂五次del Pezzo曲面在足够大的q下满足Peyre的全高度方法，同时建立了低次数分裂del Pezzo曲面上有理曲线计数函数的下界。
Integral points on singular del Pezzo surfaces
本文通过分类弱 del Pezzo 对，研究有界对数反典范高度整点，并在 $\mathbf{A}_1+\mathbf{A}_3$ 型四次奇异 del Pezzo 曲面上证明顾及奇点与直线的 Manin 型结论。
Integral Points on a del Pezzo Surface over Imaginary Quadratic Fields
本文研究虚二次域上 $\mathbf{A}_3$ 型四次 del Pezzo 曲面的整点，通过 universal torsors 进行高度有界计数并作几何解释，证明了相对奇点和直线的整点版 Manin 猜想类比。
Hasse principle violation for algebraic families of del Pezzo surfaces of degree 4 and hyperelliptic curves of genus congruent to 1 modulo 4
本文构造了无限多个显式单参数代数族，包含次数为4的del Pezzo曲面和亏格模4余1的超椭圆曲线，证明其每个K成员都违反Hasse原则，并由Brauer-Manin障碍解释。
We classify generically transitive actions of semidirect products of an additive and a multiplicative group on the projective plane. Motivated by the program to study the distribution of rational points on del Pezzo surfaces (Manin's conjecture), we determine all (possibly singular) del Pezzo surfaces that are equivariant compactifications of homogeneous spaces for such semidirect products.
A Zariski dense exceptional set in Manin's conjecture: dimension 2
本文构造了一个次数为1的 del Pezzo 曲面，其几何例外集是 Zariski 稠密的，从而提供了特征0下维度2的原始 Manin 猜想的第一个反例，并在特定假设下证明了存在无限多个此类反例。
We prove Manin's conjecture for four singular quartic del Pezzo surfaces over imaginary quadratic number fields, using the universal torsor method.
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 cone of X (in particular, for those degrees the moduli space of morphisms has the expected dimension). The result applies to a class of generalized del Pezzo surfaces which has been intensively studied in the context of the arithmetic Manin's conjecture.
We completely solve the inverse Galois problem for del Pezzo surfaces of degree $2$ and $3$ over all finite fields.
Weak weak approximation and the Hilbert property for degree-two del Pezzo surfaces
本文证明含特定有理点的二次 del Pezzo 曲面在数域上满足弱弱逼近，在 Hilbert 域上满足 Hilbert 性质，基于单有理性和圆锥纤维结果。
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.
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.
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 $\mathbb{Q}$.
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。
Torsors of Generalized del Pezzo Tori and Brauer Groups
利用上同调 Mackey 函子，给出广义 del Pezzo 簇上环面 torsor 的显式 Brauer 群分类，推广了 Blunk 对六次 del Pezzo 曲面的 Brauer 理论描述。
Let $S$ be a smooth cubic surface over a finite field $\mathbb F_q$. It is known that $\#S(\mathbb F_q) = 1 + aq + q^2$ for some $a \in \{-2,-1,0,1,2,3,4,5,7\}$. Serre has asked which values of a can arise for a given $q$. Building on special cases treated by Swinnerton-Dyer, we give a complete answer to this question. We also answer the analogous question for other del Pezzo surfaces, and consider the inverse Galois problem for del Pezzo surfaces over finite fields. Finally we give a corrected version of Manin's and Swinnerton-Dyer's tables on cubic surfaces over finite fields.
In this article, we obtain an upper bound for the number of integral points on the del Pezzo surfaces of degree two.
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.
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