Anthony Várilly-Alvarado
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.
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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 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.
We completely solve the inverse Galois problem for del Pezzo surfaces of degree $2$ and $3$ over all finite fields.
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.
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。
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 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.
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.
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.
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。
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。
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 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.
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…
We study the algebraic Brauer classes on open del Pezzo surfaces of degree $4$. I.e., on the complements of geometrically irreducible hyperplane sections of del Pezzo surfaces of degree $4$. We show that the $2$-torsion part is generated by classes of two different types. Moreover, there are two types of $4$-torsion classes. For each type, we discuss methods for the evaluation of such a class at a rational point over a $p$-adic field.
Let $X$ be a minimal del Pezzo surface of degree $2$ over a finite field $\mathbb{F}_q$. The image $Γ$ of the Galois group $\operatorname{Gal}(\overline{\mathbb{F}}_q / \mathbb{F}_q)$ in the group $\operatorname{Aut}(\operatorname{Pic}(\overline{X}))$ is a cyclic subgroup of the Weyl group $W(E_7)$. There are $60$ conjugacy classes of cyclic subgroups in $W(E_7)$ and $18$ of them correspond to minimal del Pezzo surfaces. In this paper we study which possibilities of these subgroups for minimal del Pezzo surfaces of degree $2$ can be achieved for given $q$.
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。
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。
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.
In this article, we obtain an upper bound for the number of integral points on the del Pezzo surfaces of degree two.
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.
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.
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。
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.
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