Fok-Shuen Leung
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 for a particular surface.
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Classification of curves in a projective space occupies minds of many mathematicians. First step in doing so is classification of curves on a given surface. This brings us to consideration of the nonsingular Del Pezzo Surface in $P^4_k.$ We describe conditions for a divisor to be very ample and condition for a divisor class to contain an irreducible curve.
We prove a criterion of nonsingularity of a complete intersection of two fiberwise quadrics in a scroll over $P^1$. As a corollary we derive the following addition to the Alexeev theorem on rationality of standard Del Pezzo fibrations of degree 4 over $P^1$: we prove that any fibration of this kind with the topological Euler characteristic -4 is rational.
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 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.
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 ${\mathbb Q}$.
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 Lemma which yields an inductive approach to classifying relatively free sections for a del Pezzo fibration over a curve. Using this lemma we prove Geometric Manin's Conjecture for certain split del Pezzo surfaces of degree $\geq 2$ admitting a birational morphism to $\mathbb P^2$ over the ground field.
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.
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 0. We also investigate the principles of Geometric Manin's Conjecture for weak del Pezzo surfaces. In the course of this investigation, we give examples of weak del Pezzo surfaces defined over $\mathbb{F}_{2}(t)$ or $\mathbb{F}_{3}(t)$ such that the exceptional sets in Manin's Conjecture are Zariski dense.
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。
In this paper, we study rigidity of nonsingular del Pezzo fibrations over a germ of smooth curve.
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。
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 the irreducibility of moduli spaces of rational curves on a general del Pezzo threefold of Picard rank $1$ and degree $1$. As corollaries, we confirm Geometric Manin's conjecture and enumerativity of certain Gromov-Witten invariants for these threefolds.
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)
We develop a strategy to classify the components of the space of sections of a del Pezzo fibration over $\mathbb{P}^{1}$. In particular, we prove the Movable Bend and Break lemma for del Pezzo fibrations. Our approach is motivated by Geometric Manin's Conjecture and proves upper bounds on the associated counting function. We also give applications to enumerativity of Gromov-Witten invariants and to the study of the Abel-Jacobi map.
We classify all generalized del Pezzo surfaces (i.e., minimal desingularizations of singular del Pezzo surfaces containing only rational double points) whose universal torsors are open subsets of hypersurfaces in affine space. Equivalently, their Cox rings are polynomial rings with exactly one relation. For all 30 types with this property, we describe the Cox rings in detail. These explicit descriptions can be applied to study Manin's conjecture on the asymptotic behavior of the number of rational points of bounded height for singular del Pezzo surfaces, using the universal torsor method.
We prove that every del Pezzo surface of degree two over a finite field is unirational, building on the work of Manin and an extension by Salgado, Testa, and Várilly-Alvarado, who had proved this for all but three surfaces. Over general fields of characteristic not equal to two, we state sufficient conditions for a del Pezzo surface of degree two to be unirational.
The irreducibility of the spaces of rational curves on del Pezzo manifolds
本文通过研究赋值映射纤维，证明 Picard 秩 1、维数至少 4 的 del Pezzo 流形上有理曲线空间不可约，并由此推出几何 Manin 猜想。
Among geometrically rational surfaces, del Pezzo surfaces of degree two over a field k containing at least one point are arguably the simplest that are not known to be unirational over k. Looking for k-rational curves on these surfaces, we extend some earlier work of Manin on this subject. We then focus on the case where k is a finite field, where we show that all except possibly three explicit del Pezzo surfaces of degree two are unirational over k.
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。
We prove that every del Pezzo surface of degree two over a finite field is unirational, building on the work of Manin and an extension by Salgado, Testa, and Várilly-Alvarado, who had proved this for all but three surfaces. Over general fields, we state several sufficient conditions for a del Pezzo surface of degree two to be unirational.
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.
In this paper, we propose a feasible algorithm to give an explicit basis of the space of regular differential forms on the nonsingular projective model of any given plane algebraic curve. The algorithm is demonstrated for concrete examples, with our implementation over the computer algebra system Magma. As an application, we also describe the Cartier-Manin matrix of the nonsingular projective curve with respect to the basis computed by the algorithm.
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。
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。
Moduli space of genus one curves on quartic and quintic del Pezzo threefolds
本文研究四次和五次 del Pezzo 三维簇上光滑亏格一曲线的空间，描述了 Kontsevich 模空间的不可约分支并分类了从一般椭圆曲线出发的态射空间，验证了复数域上所有四次和五次 del Pezzo 三维簇的几何 Manin 猜想。
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 compute a naturally defined measure of the size of the nef cone of a Del Pezzo surface. The resulting number appears in a conjecture of Manin on the asymptotic behavior of the number of rational points of bounded height on the surface. The nef cone volume of a Del Pezzo surface Y with (-2)-curves defined over an algebraically closed field is equal to the nef cone volume of a smooth Del Pezzo surface of the same degree divided by the order of the Weyl group of a simply-laced root system associated to the configuration of (-2)-curves on Y. When Y is defined over a non-closed field of characte…
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.
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。
Rational curves on Fano threefolds with Gorenstein terminal singularities
本文研究具有Gorenstein终端奇点的Fano三维簇上的有理曲线，推广了几何Manin猜想至奇异情形，并证明了某些奇异del Pezzo三维簇的该猜想。
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…
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 projective homogeneous space of the semisimple group with the same root system, equivariantly with respect to the maximal torus action. We prove this conjecture for del Pezzo surfaces of degrees greater than 1. Our proof uses an inductive process based on representations of Lie algebras corresponding to Hermitian symmetric pairs.
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。
We prove the existence of Kahler-Einstein metrics on a nonsingular section of the Grassmannian $\mathrm{Gr}(2, 5)\subset\mathbb{P}^9$ by a linear subspace of codimension 3, and the Fermat hypersurface of degree 6 in $\mathbb{P}(1,1,1,2,3)$. We also show that a global log canonical threshold of the Mukai--Umemura variety is equal to 1/2.
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。
We explain a classical construction of a del Pezzo surface of degree d = 4 or 5 as a smooth order two congruence of lines in 3-space whose focal surface is a quartic surface $X_{20-d}$ with 20-d ordinary double points. We also show that $X_{15}$ can be realized as a hyperplane section of the Castelnuovo-Richmond-Igusa quartic hypersurface. This leads to the proof of rationality of the moduli space of 15-nodal quartic surfaces. We discuss some other birational models of $X_{15}$: quartic symmetroids, 5-nodal quartic surfaces, 10-nodal sextic surfaces in $P^4$ and nonsingular surfaces of degree …
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。
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