Michel Brion, Emmanuel Peyre
We say that a smooth algebraic group $G$ over a field $k$ is very special if for any field extension $K/k$, every $G_K$-homogeneous $K$-variety has a $K$-rational point. It is known that every split solvable linear algebraic group is very special. In this note, we show that the converse holds, and discuss its relationship with the birational classification of algebraic group actions.
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This paper contains a survey of recent developments in investigation of word equations in simple matrix groups and polynomial equations in simple (associative and Lie) matrix algebras along with some new results on the image of word maps on algebraic groups defined over special fields: complex, real, p-adic (or close to such), or finite.
Given a real curve, we study special linear systems called "very special" for which the dimension does not satisfy a Clifford type inequality. We classify all these very special linear systems when the gonality of the curve is small.
We study special linear systems called "very special" whose dimension does not satisfy a Clifford type inequality given by Huisman. We classify all these very special linear systems when they are compounded of an involution. Examples of very special linear systems that are simple are also given.
New examples of geometrically special varieties: K3 surfaces, Enriques surfaces, and algebraic groups
本文验证椭圆K3曲面和代数群是Javanpeykar-Rousseau意义下的几何特殊簇，并证明附加假设下移除余维≥2闭子集后该特殊性仍保持，提供了几何特殊簇的新例子。
This note is about a $16$-dimensional family of surfaces of general type with $p_g=2$ and $q=0$ and $K^2=1$, called "special Horikawa surfaces". These surfaces, studied by Pearlstein-Zhang and by Garbagnati, are related to K3 surfaces. We show that special Horikawa surfaces have a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. As a consequence, the Chow ring of special Horikawa surfaces displays K3-like behaviour.
We present a new notion of speciality which is valid for Bohr - Sommerfeld lagrangian submanifolds. For algebraic varieties it leads to the construction of finite dimensional moduli spaces which are algebraic starting with any ample line bundle. The construction can be exploited both in Geometric Quantization and Mirror Symmetry.
Let S be a smooth algebraic surface satisfying the following property: H^i(\oc_S(C))=0 (i=1,2) for any irreducible and reduced curve C of S. The aim of this paper is to provide a characterization of special linear systems on S which are singular along a set of double points in general position. As an application, the dimension of such systems is evaluated in case S is an Abelian, an Enriques, a K3 or an anticanonical rational surface.
Motivated by the Beauville-Voisin conjecture about Chow rings of powers of $K3$ surfaces, we consider a similar conjecture for Chow rings of powers of EPW sextics. We prove part of this conjecture for the very special EPW sextic studied by Donten-Bury et alii. We also prove some other results concerning the Chow groups of this very special EPW sextic, and of certain related hyperkähler fourfolds.
Informed by the Bloch-Beilinson conjectures, Voisin has made a conjecture about $0$-cycles on self-products of Calabi-Yau varieties. In this note, we consider variant versions of Voisin's conjecture for cubic fourfolds, and for hyperkähler varieties. We present examples for which these conjectures are verified, by considering certain very special cubic fourfolds and their Fano varieties of lines.
We devise a new criterion for linear independence over function fields. Using this tool in the setting of dual t-motives, we find that all algebraic relations among special values of the geometric function field Gamma-function are explained by the standard functional equations.
What makes an algebraic curve special?
本文综述了代数曲线及模空间 $\mathcal{M}_g$ 中特殊子簇的多种“特殊”含义，并结合具体例子重点介绍了 Hodge 理论与 Teichmüller 理论中非典型交点的双代数视角。
Special triple covers of algebraic surfaces
本文研究代数曲面的特殊三重覆盖，通过刻画 Tschirnhausen 丛为分裂秩三向量丛的商，给出存在性的充要条件，并完成特殊三重平面的分类，发现两个 K3 曲面族。
Let G be an algebraic group defined over an algebraically closed field k of characteristic zero. We give a simple proof of the following result: if H^1(L, G) = {1} for some finitely generated field extension L/k of transcendence degree \ge 3 then H^1(K, G) = {1} for every field extension K/k.
On the non-special divisors in algebraic function fields defined over $\mathbb{F}_q$
本文综述了有限域上代数函数域中非特殊除子的研究成果，并探讨了缺陷为k的曲线中此类除子的存在性问题。
The special geometry of calibrated cycles, closely related to mirror symmetry among Calabi--Yau 3-folds, is itself a real form of a new subject, which we call slightly deformed algebraic geometry. On the other hand, both of these geometries are parallel to classical gauge theories and their complexifications. This article explains this parallelism, so that the appearance of invariants of new type in complexified gauge theory (see Donaldson--Thomas [D-T] and Thomas [T]) can be accompanied by analogous invariants in the theory of special Lagrangian cycles, for which the development is at present…
The p-cohomology of an algebraic variety in characteristic p lies naturally in the category $D_{c}^{b}(R)$ of coherent complexes of graded modules over the Raynaud ring (Ekedahl-Illusie-Raynaud). We study homological algebra in this category. When the base field is finite, our results provide relations between the the absolute cohomology groups of algebraic varieties, log varieties, algebraic stacks, etc. and the special values of their zeta functions. These results provide compelling evidence that $D_{c}^{b}(R)$ is the correct target for p-cohomology in characteristic p.
This paper is devoted to the investigation of selected situations when the computation of projective (and other) equivalences of algebraic varieties can be efficiently solved with the help of finding projective equivalences of finite sets on the projective line. In particular, we design a unifying approach that finds for two algebraic varieties $X,Y$ from special classes an associated set of automorphisms of the projective line (the so called good candidate set) consisting of candidates for the construction of possible mappings $X\rightarrow Y$. The functionality of the designed method is pres…
We study the degrees of generators of the ideal of a projected Veronese variety $v_2(\mathbb{P}^3)\subset \mathbb{P}^9$ to $\mathbb{P}^6$ depending on the center of projection. This is related to the geometry of zero dimensional schemes of length $8$ in $\mathbb{A}^4$, Cremona transforms of $\mathbb{P}^6$, and the geometry of Tonoli Calabi-Yau threefolds of degree $17$ in $\mathbb{P}^6$.
We reconstruct derived Witt groups via special linear algebraic cobordism. There is a morphism of ring cohomology theories which sends the canonical Thom class in special linear cobordism to the Thom class in the derived Witt groups. We show that for every smooth variety X this morphism induces an isomorphism between MSL^{*,*}(X)[h^{-1}] with the "extended" coefficient ring MSL^{4*,2*}(pt) -> W^{2*}(pt) and Laurent polynomial ring over the derived Witt groups W^*(X), where h is the stable Hopf map. This result is an analogue of the result by Panin and Walter reconstructing hermitian K-theory u…
This note contains some examples of hyperkähler varieties $X$ having a group $G$ of non-symplectic automorphisms, and such that the action of $G$ on certain Chow groups of $X$ is as predicted by Bloch's conjecture. The examples range in dimension from $6$ to $132$. For each example, the quotient $Y=X/G$ is a Calabi-Yau variety which has interesting Chow-theoretic properties; in particular, the variety $Y$ satisfies (part of) a strong version of the Beauville-Voisin conjecture.
We study the existence of non-special divisors of degree $g$ and $g-1$ for algebraic function fields of genus $g\geq 1$ defined over a finite field $\F_q$. In particular, we prove that there always exists an effective non-special divisor of degree $g\geq 2$ if $q\geq 3$ and that there always exists a non-special divisor of degree $g-1\geq 1$ if $q\geq 4$. We use our results to improve upper and upper asymptotic bounds on the bilinear complexity of the multiplication in any extension $\F_{q^n}$ of $\F_q$, when $q=2^r\geq 16$.
We prove some general density statements about the subgroup of invertible points on intermediate jacobians; namely those points in the Abel-Jacobi image of nullhomologous algebraic cycles on projective algebraic manifolds.
In this paper we demonstrate that the notion of inflection points and extactic points on plane algebraic curves can be suitably transferred to curves in $\mathbb{P}^1\times \mathbb{P}^1$. More precisely, we describe osculating curves and study Weierstrass points of algebraic curves in the surface $\mathbb{P}^1\times \mathbb{P}^1$ with respect to certain linear systems. In particular, we study points where a fiber of $\mathbb{P}^1\times \mathbb{P}^1$ is tangent, and points with a hyperosculating $(1,1)$-curve. In the first case we find Hessian-like curves that intersect the curve in these point…
Regular algebraic surfaces isogenous to a higher product of curves can be obtained from finite groups with ramification structures. We find unmixed ramification structures for finite groups constructed as p-quotients of particular infinite groups with special presentations related to finite projective planes.
Algebraic hyperbolicity of very general hypersurfaces in weighted projective spaces
本文研究加权射影空间中非常一般超曲面的代数双曲性，通过给出丰沛线丛倍数 $m$ 的界，证明了相应零点轨迹具有代数双曲性。
We prove that a curve of degree $dk$ on a very general surface of degree $d \geq 5$ in $\mathbb{P}^3$ has geometric genus at least $\frac{dk(d-5)+k}{2} + 1$. This improves bounds given by G. Xu. As a corollary, we conclude that the very general quintic surface in $\mathbb{P}^3$ is algebraically hyperbolic.
Algebraic hyperbolicity of very general hypersurfaces in projective spaces
本文证明了射影空间中非常一般超曲面代数双曲性的最后未解情形，即四次三维簇，完成了分类，方法基于Coskun--Riedl构造及稳定映射模空间和对数几何。
A linear algebraic group G defined over a field k is called special if every G-torsor over every field extension of k is trivial. In 1958 Grothendieck classified special groups in the case where the base field is algebraically closed. In this paper we describe the derived subgroup and the coradical of a special reductive group over an arbitrary field k. We also classify special semisimple groups, special reductive groups of inner type and special quasisplit reductive groups over an arbitrary field k.
Slices of the special linear algebraic cobordism spectrum
本文计算了指数特征为e的域F上特殊线性代数配边谱MSL在e可逆化后的切片，结果用特殊酉配边谱的Adams-Novikov谱序列第二页表示，并应用于MSL的切片谱序列以确定其同伦群的前几个Milnor-Witt茎，同时建立了有理特殊线性代数配边谱在任意qcqs概形上的分解。
Spectral Fingerprints of Algebraic Cycles: A Hodge-Theoretic Approach to the Hodge Conjecture and Special L-Values
本文提出检测复代数簇上代数闭链的“谱指纹理念”，通过霍奇理论与算术数据关联闭链代数性，证明Kummer K3曲面新除子的存在，为霍奇猜想研究提供新视角。
Some remark on real algebraic maps which are topologically special generic maps and generalize the canonical projections of the unit spheres
本文针对推广单位球面典型投影的拓扑特殊一般映射，构造了像为光滑嵌入流形的实代数映射，作为先前构造的补充注记。
We study the algebraic hyperbolicity of very general hypersurfaces in $\mathbb{P}^m \times \mathbb{P}^n$ by using three techniques that build on past work by Ein, Voisin, Pacienza, Coskun and Riedl, and others. As a result, we completely answer the question of whether or not a very general hypersurface of bidegree $(a,b)$ in $\mathbb{P}^m \times \mathbb{P}^n$ is algebraically hyperbolic, except in $\mathbb{P}^3 \times \mathbb{P}^1$ for the bidegrees $(a,b)= (7,3), (6,3)$ and $(5,b)$ with $b\geq 3.$ As another application of these techniques, we improve the known result that very general hypers…
Algebraic hyperbolicity of very general hypersurfaces in homogeneous varieties
本文推广了Coskun、Riedl和Yeong的技巧，得到了有理齐次簇中非常一般超曲面的代数双曲性的几乎最优次数界，并给出了格拉斯曼簇、正交/辛格拉斯曼簇及旗簇等实例。
Let Z be an affine algebraic variety and ED(Z)= max(2 dim Z+1, dim TZ). Let X be a smooth algebraic variety isomorphic to a semi-simple linear algebraic group whose Lie algebra is a sum of special linear Lie algebras. We show that if dim X > ED(Z) -1, then Z admits a closed embedding into X. We also show that for every smooth affine flexible variety Y there is a closed embedding of $Z$ into the the product of Y and an affine n-space provided that n > dim Z- 2 and dim Y +n > ED (Z) -1.
Algebraic Independence of Special Points on Shimura Varieties
本文利用Shimura簇与代数群间的对应证明特殊点像的代数独立性，统一乘法与线性独立性结果，并应用于奇异模和Abel簇。
In this first work dedicated to the generalisation of classic algebraic geometry to non algebraically closed fields and axiomatisable classes of fields, we develop the foundations for equiresidual algebraic geometry (EQAG), i.e. algebraic geometry over any commutative field $k$, algebraically closed or not. This is possible thanks to the existence in non algebraically closed fields of many normic forms, i.e. homogeneous polynomials with no non-trivial zero rational in $k$, and relies on an equiresidual generalisation of Hilbert's Nullstellensatz and the Jacobson radical in finitely presented $…
The unit map of the algebraic special linear cobordism spectrum
本文利用框架对应技术计算动机Thom谱的无限$\mathbb{P}^1$-环空间，并证明代数特殊线性配边谱的单位映射在$\mathbb{G}_m$-同伦层上诱导同构。
Specialization and rigidity
本文描述了由Ofer Gabber创立的通用方法，用于证明族中的非常一般纤维具有特定性质，并通过代数动力学、Galois理论等领域的具体例子加以说明。
Recently, Haase and Ilten initiated the study of classifying algebraically hyperbolic surfaces in toric threefolds. We complete this classification for $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$, $\mathbb{P}^2 \times \mathbb{P}^1$, $\mathbb{F}_e \times \mathbb{P}^1$ and the blowup of $\mathbb{P}^3$ at a point, augmenting our earlier work on $\mathbb{P}^3$. In the process, we codify several different techniques for proving algebraic hyperbolicity, allowing us to prove similar results for hypersurface in any variety admitting a group action with dense orbit.
This survey paper is devoted to Riemannian manifolds with special holonomy. To any Riemannian manifold of dimension n is associated a closed subgroup of SO(n), the holonomy group; this is one of the most basic invariants of the metric. A famous theorem of Berger gives a complete (and rather small) list of the groups which can appear. Surprisingly, the compact manifolds with holonomy smaller than SO(n) are all related in some way to Algebraic Geometry. This leads to the study of special algebraic varieties (Calabi-Yau, complex symplectic or complex contact manifolds) for which Riemannian geomet…
In the previous paper by Asakura-Otsubo-Terasoma, we prove that the special values of the hypergeometric function 3F2 at 1 are linear combinations of logarithms of algebraic numbers and 1 over algebraic numbers, if exponents are rational numbers satisfying a certain arithmetic condition. Aoki and Shioda completely classified these sets of rational numbers satisfying this condition in connection with Hodge cycles on Fermat surfaces. In this paper, we give an explicit expression of special values of hypergoemetricy 3F2 which does not belong to exceptional characters.
In this note, we show that a very general system of algebraic linear partial differential equations has zero kernel, applying basic techniques of the theory of jet-modules and elementary base change theory. In particular, in contrast to the differentiable case, a very general system of algebraic ordinary differential equations has zero kernel.
In this article we study interpolation estimates on a special class of compactifications of commutative algebraic groups constructed by Serre. We obtain a large quantitative improvement over previous results due to Masser and the first author and our main result has the same level of accuracy as the best known multiplicity estimates. The improvements come both from using special properties of the compactifications which we consider and from a different approach based upon Seshadri constants and vanishing theorems.
A broadly applicable geometric approach for constructing nef divisors on blow ups of algebraic surfaces at n general points is given; it works for all surfaces in all characteristics for any n. This construction is used to obtain substantial improvements for currently known lower bounds for n point Seshadri constants. Remarks are included about a range of applications to classical problems involving linear systems on P2.
Explicit non-special divisors of small degree, algebraic geometric hulls, and LCD codes from Kummer extensions
本文通过Kummer扩张的有理位和Weierstrass半群显式构造非特殊因子，得到指定包络维数的代数几何码及LCD码，并回答了Ballet-Le Brigand的开放问题。
Adapting focal loci techniques used by Chiantini and Lopez, we provide lower bounds on the genera of curves contained in very general surfaces in Gorenstein toric threefolds. We illustrate the utility of these bounds by obtaining results on algebraic hyperbolicity of very general surfaces in toric threefolds.
We introduce a simplified version of the Grothendieck group of algebraic varieties and use it to show that birational types specialize in families with mild singularities of the central fiber.
An interpolation between special linear and general algebraic cobordism $\text{MSL}$ and $\text{MGL}$
本文在稳定动机同伦范畴中建立了 $\text{MSL}$ 与 $\text{MGL}$ 之间的插值关系，由此得到 $\text{MGL}$ 的滤过，并计算了特征零域上 $\text{MSL}$ 的第一切片和稳定同伦群的第一行。
Let N be a prime and let A be a quotient of J_0(N) over Q associated to a newform such that the special L-value of A (at s=1) is non-zero. Suppose that the algebraic part of the special L-value of A is divisible by an odd prime q such that q does not divide the numerator of (N-1)/12. Then the Birch and Swinnerton-Dyer conjecture predicts that the q-adic valuations of the algebraic part of the special L-value of A and of the order of the Shafarevich-Tate group are both positive even numbers. Under a certain mod q non-vanishing hypothesis on special L-values of twists of A, we show that the q-ad…
The algebraic Green-Griffiths-Lang conjecture for complements of very general pairs of divisors
本文证明了 $\mathbb{P}^n$ 中总次数为 $2n$ 的非常一般除子对之补集在模去真闭子簇后代数双曲，从而为 Lang-Vojta 与 Green-Griffiths 猜想提供新证据，并部分推广了已有结果。
In this article we study the transitivity of the group of automorphisms of real algebraic surfaces. We characterize real algebraic surfaces with very transitive automorphism groups. We give applications to the classification of real algebraic models of compact surfaces: these applications yield new insight into the geometry of the real locus, proving several surprising facts on this geometry. This geometry can be thought of as a half-way point between the biregular and birational geometries.
Tropical algebraic geometry is an active new field of mathematics that establishes and studies some very general principles to translate algebro-geometric problems into purely combinatorial ones. This expository paper gives an introduction to these new techniques with a special emphasis on the recent applications to problems in enumerative geometry.
Given a connected algebraic group G over an algebraically closed field and a G-homogeneous space X, we describe the Chow ring of G and the rational Chow ring of X, with special attention to the Picard group. Also, we investigate the Albanese and the "anti-affine" fibrations of G and X.
In the first part of this paper we consider compact algebraic manifolds M^2n with an algebraic (n-1)-Torus action. We show that there is a T-invariant meromorphic section $σ$ of the canonical bundle of M. Any such $σ$ defines a divisor D. On the complement M'=M-D we have a trivialization of the canonical bundle and a T-action. If the first cohomology of M' vanishes, then results of [2] imply that there is a Special Lagrangian (SLag) fibration on M'. We study how the fibers compactify in M and give examples of SLag fibrations on M', including some cases there the first cohomology of M' doesn't …
Special Bohr - Sommerfeld geometry, first formulated for simply connected symplectic manifolds (or for simple connected algebraic varieties), gives rise to some natural problems for the simplest example in non simply connected case. Namely for any algebraic curve one can define a correspondence between holomorphic differentials and certain finite graphs. Here we ask some natural questions appear with this correspondence. It is a partial answer to the question of A. Varchenko about possibility of applications of Special Bohr -Sommerfeld geometry in non simply connected case. The russian version…
In this paper we discuss the geometry of affine Deligne Lusztig varieties with very special level structure, determining their dimension and connected and irreducible components. As application, we prove the Grothendieck conjecture for Shimura varieties with very special level at $p$ and a function field analogue.
The geometric diagonal of the special linear algebraic cobordism
本文构造了c1-球面配边谱的动机版本，通过建立与其他动机Thom谱的联系，计算了局部Dedekind域上特殊线性代数配边同伦群的P1-对角，并讨论了动机Hopf元素η的作用、2-局部化及2-初级挠子群，给出了完全答案。
Sets of Special Subvarieties of Bounded Degree
本文针对代数簇族的 Hodge 轨迹，提出计算给定次数内全部弱特殊子簇的算法，并据此证明特殊与弱特殊子簇有界次数集合的有限性猜想。
Let $k$ be a base field and $G$ be an algebraic group over $k$. J.-P. Serre defined $G$ to be special if every $G$-torsor $T \to X$ is locally trivial in the Zariski topology for every reduced algebraic variety $X$ defined over $k$. In recent papers an a priori weaker condition is used: $G$ is called special if every $G$-torsor $T \to Spec(K)$ is split for every field $K$ containing $k$. We show that these two definitions are equivalent. We also generalize this fact and propose a strengthened version of the Grothendieck-Serre conjecture based on the notion of essential dimension.
Degenerations, contractions and deformations of various algebraic structures play an important role in mathematics and physics. There are many different definitions and special cases of these notions. We try to give a general definition which unifies these notions and shows the connections among them. Here we focus on contractions of Lie algebras and algebraic groups.
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