Louis Esser
Unlike the classical Brauer group of a field, the Brauer-Grothendieck group of a singular scheme need not be torsion. We show that there exist integral normal projective surfaces over a large field of positive characteristic with non-torsion Brauer group. In contrast, we demonstrate that such examples cannot exist over the algebraic closure of a finite field.
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Let $S$ be a scheme and let $π: \mathcal{G} \to S$ be a $\mathbb{G}_{m,S}$-gerbe corresponding to a torsion class $[\mathcal{G}]$ in the cohomological Brauer group $\mathrm{Br}'(S)$ of $S$. We show that the cohomological Brauer group $\mathrm{Br}'(\mathcal{G})$ of $\mathcal{G}$ is isomorphic to the quotient of $\mathrm{Br}'(S)$ by the subgroup generated by the class $[\mathcal{G}]$. This is analogous to a theorem proved by Gabber for Brauer-Severi schemes.
We give an algorithm to explicitly determine all elements of the $q$-torsion (for $q$ an odd prime) of the Brauer group of an elliptic curve over any base field of characteristic different from $q$, containing a primitive $q$-th root of unity. These elements of the Brauer group are given as tensor products of symbol algebras over the function field of the elliptic curve. We give sufficient conditions to determine if the Brauer classes that arise are trivial. Using our algorithm, we derive an upper bound on the symbol length of the prime torsion of $\mathrm{Br}(E)/\mathrm{Br}(k)$.
Boundedness of the p-primary torsion of the Brauer groups of K3 surfaces
本研究针对有限生成域上K3曲面的超越布劳尔群展开，证明其通常有限，仅在域为正特征p且曲面超奇异时被p零化，完善了相关代数几何理论。
Let U^{'s}_{L}(n,d) be the moduli space of stable vector bundles of rank $n$ and fixed determinant L of degree d on a nodal curve Y. The moduli space of semistable vector bundles of rank n and degree $d$ will be denoted by U'_Y(n,d). We calculate the Brauer groups of U^{'s}_{L}(n,d)$. We study the question of rationality of $U^{'s}_{L}(n,d)$ and $U'_Y(n,d)$.
In this paper, we generalize a result of Karpenko on the torsion in the second quotient of the gamma filtration for Severi-Brauer varieties to higher degrees. As an application, we provide a nontrivial torsion in higher Chow groups and the topological filtration of the associated generic variety and obtain new upper bounds for the annihilators of the torsion subgroups in the Chow groups of a large class of Severi-Brauer varieties. In particular, using the torsion in higher degrees, we show indecomposability of certain algebras.
We discuss the geometry of the genus one fibrations associated to an elliptic fibration on a K3 surface. We show that the two-torsion subgroup of the Brauer group of a general elliptic fibration is naturally isomorphic to the two-torsion of the Jacobian of a curve associated to the fibration. We remark that this is related to Recillas' trigonal construction. Finally we discuss the two-torsion in the Brauer group of a general K3 surface with a polarisation of degree two.
We relate the Brauer group of a Kummer surface to the Brauer group of the corresponding abelian surface. For many pairs of elliptic curves over the rational numbers we prove that the Kummer surface attached to their product has trivial Brauer group.
p-Primary Torsion of the Brauer Group in Characteristic p
研究特征p>0的代数闭域上光滑完备簇的Brauer群的p-主分量，利用平展上同调与de Rham-Witt复形建立计算框架。
We use recent advances in the local evaluation of Brauer elements to study the role played by {\it odd} torsion elements of the Brauer group in the arithmetic of diagonal quartic surfaces over {\it arbitrary} number fields. We show that over a local field if the order of the Brauer element is odd and coprime to the residue characteristic then the evaluation map it induces on the local points is constant. Over number fields we give a sufficient condition on the coefficients of the equation, which is mild and easy to check, so that the odd torsion does not obstruct weak approximation. We also no…
We show that the torsion in the group of indecomposable $(2,1)$-cycles on a smooth projective variety over an algebraically closed field is isomorphic to a twist of its Brauer group, away from the characteristic. In particular, this group is infinite as soon as $b_2-ρ>0$. We derive a new insight into Roitman's theorem on torsion $0$-cycles over a surface.
Boundedness of the p-primary torsion of the Brauer group of products of varieties
本研究证明域上簇乘积的Brauer群商群有有限指数，给出阿贝尔簇Brauer群p-准素挠有界性的直接证法，并解答了Kummer曲面超越Brauer群相关的公开问题。
En utilisant un théorème de Gabber sur les altérations, on démontre un résultat décrivant la partie de torsion première à $p$ du groupe de Brauer non ramifié d'une variété $V$ lisse et géométriquement intègre sur un corps global de caractéristique $p$ au moyen de l'évaluation des éléments de $\Br V$ sur ses points locaux. Using a theorem of Gabber on alterations, we prove a result describing the prime-to-$p$ torsion part of the unramified Brauer group of a smooth and geometrically integral variety $V$ over a global field of characteristic $p$ by evaluating the elements of $\Br V$ at its local …
We construct unramified central simple algebras representing 2-torsion classes in the Brauer group of a hyperelliptic curve, and show that every 2-torsion class can be constructed this way when the curve has a rational Weierstrass point or when the base field is C_1. In general, we show that a large (but in general proper) subgroup of the 2-torsion classes are given by the construction. Examples demonstrating applications to the arithmetic of hyperelliptic curves defined over number fields are given.
Let $A$ be an abelian variety over a field finitely generated over $\mathbb{Q}$. We show that the finiteness of the $\ell$-primary torsion subgroup of the higher Brauer group is a sufficient criterion for the Tate conjecture to hold. Furthermore, we extend methods for computations of transcendental Brauer groups to higher Brauer groups.
Transcendental Brauer elements are notoriously difficult to compute. Work of Wittenberg, and later, Ieronymou, gives a method for computing 2-torsion transcendental classes on surfaces that have a genus 1 fibration with rational 2-torsion in the Jacobian fibration. We use ideas from a descent paper of Poonen and Schaefer to remove this assumption on the rational 2-torsion.
For a smooth and projective variety over a number field with torsion free geometric Picard group and finite transcendental Brauer group we show that only the archimedean places, the primes of bad reduction and the primes dividing the order of the transcendental Brauer group can turn up in the description of the Brauer-Manin set.
The class in the Brauer group of a quaternion algebra over a field is 2-torsion. We study the following question: Which 2-torsion elements of the Brauer group of a complex function field are representable by quaternion algebras? Using intersection theory to show that a certain cohomology class (on a smooth projective model) is the class of an algebraic cycle, we arrive at an obstruction, defined on a subgroup of the 2-torsion of the Brauer group, to representability by quaternion algebras. For the function fields of some complex threefolds, the obstruction map is computed and found to be nontr…
Boundedness of the $p$-primary torsion of the Brauer group of an abelian variety
本文证明特征p>0的有限生成域上阿贝尔簇的超验Brauer群的p∞-挠有界，回答了Skorobogatov–Zarhin问题的变体，并借助除子的平坦Tate猜想，同时阐明其他几何Galois不变挠类可无限p-可除且与Néron–Severi群特殊化映射非满射相关。
A Brauer pair is a pair (X, α) where X is a quasi-projective variety over an algebraically closed field and α is an element in the 2-torsion part of the Brauer group of the function field of X. A Brauer pair (Y, α) is a terminal pair if the Brauer discrepancy of (Y, α) is positive. We show that given a Brauer pair (X, α), there is a terminal pair (Y, α) with a birational morphism Y -> X. In short, any Brauer pair admits a terminal resolution.
On the topological Brauer group of generalized Kummer varieties
本文研究了广义Kummer变种的拓扑Brauer群，证明其在特定维数下消失或为8- torsion。
We prove that a smooth proper universally CH_0-trivial variety X over a field k has universally trivial Brauer group. This fills a gap in the literature concerning the p-torsion of the Brauer group when k has characteristic p.
A natural question is to determine which algebraic stacks are qoutient stacks. In this paper we give some partial answers and relate it to the old question of whether, for a scheme X, the natural map from the Brauer goup (equivalence classes of Azumaya algebras) to the cohomological Brauer group (the torsion subgroup of $H^2(X,{\mathbb G}_m)$ is surjective.
On p-torsions of geometric Brauer groups
本文在正特征有限生成域上的光滑射影簇上，证明了 $p$-初级 Brauer 群指数有限性等价于除子的 Tate 猜想，并利用刚性上同调 Leray 谱序列证明了特定映射余核的指数有限性，从而完成了 Artin-Grothendieck 定理向更高相对维数推广的 $p$-初级部分。
Brauer groups of abelian varieties over fields of finite characteristic
本文研究了有限特征域上阿贝尔簇的布雷厄姆群，重点分析了 p-原torsion 部分的结构，并通过同源类分类和埃克达尔-奥尔特类型计算了维数。
Zero cycles on Severi--Brauer flag varieties
本文研究了塞弗里-布拉uer旗形 varieties 上的零次循环 Chow 群，证明了其在特定条件下为 (d, n/d)-torsion，并给出了新的实例和更严格的 torsion 绑定。
The Brauer groups of moduli of genus three curves, abelian threefolds and plane curves
本文通过计算低阶上同调不变量，确定了特征不为2的域上亏格3曲线模栈、特定条件下平面曲线模栈及3维主极化阿贝尔簇模栈的Brauer群的$\ell$-准原挠。
We provide explicit generators for the torsion of the second cohomology of bielliptic surfaces, and we use this to study pullback map between Brauer group of a bielliptic surface and that of its canonical cover.
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.
We classify elliptic K3 surfaces in characteristic $p$ with $p^n$-torsion sections. For $p^n\geq3$ we verify conjectures of Artin and Shioda, compute the heights of their formal Brauer groups, as well as Artin invariants and Mordell--Weil groups in the supersingular cases.
We prove that the Brauer group of TMF is isomorphic to the Brauer group of the derived moduli stack of elliptic curves. Then, we compute the local Brauer group, i.e., the subgroup of the Brauer group of elements trivialized by some étale cover of the moduli stack, up to a finite 2-torsion group.
Let T be a torus (not assumed to be split) over a field F, and denote by $_n{H^{2}_{et}(X,Gm)}$ the subgroup of elements of exponent dividing n in the cohomological Brauer group of a scheme X over the field F. We provide conditions on X and n for which the pull-back homomorphism $_n{H^2_{et}(T,Gm)}\to_n{H^2_{et}(X\times T,Gm)}$ is an isomorphism. We apply this to compute the Brauer group of some reductive groups and of non singular affine quadrics. Apart from this, we investigate the p-torsion of the Azumaya algebra defined Brauer group of a regular affine scheme over a field F of characterist…
In this paper we show that the l^n-torsion part of the cohomological Brauer groups of certain schemes associated to symmetric powers of a projective smooth curve over a separably closed field k are isomorphic, when `l is invertible in k. The schemes considered are the Symmetric powers themselves, then the corresponding Picard schemes and also certain Quot-schemes. We also obtain similar results for Prym varieties associated to certain finite covers of such curves: we prove such results only for curves defined over the field of complex numbers.
In this paper we consider the Brauer groups of algebraic stacks and GIT quotients: the only algebraic stacks we consider in this paper are quotient stacks [X/G], where X is a smooth scheme of finite type over a field k, and G is a linear algebraic group over k and acting on X, as well as various moduli stacks of principal G-bundles on a smooth projective curve X, associated to a reductive group G. We also consider the Brauer groups of the corresponding coarse moduli spaces, which most often identify with the corresponding GIT-quotients. One conclusion that we seem to draw then is that the Brau…
Merkurjev's theorem--the statement that the 2-torsion of the Brauer group is represented by Clifford algebras of quadratic forms--is in general false when the base is no longer a field. The work of Parimala, Scharlau, and Sridharan proves the existence of smooth complete curves over local fields, over which Merkurjev's theorem is equivalent to the existence of a rational theta characteristic. Here, we prove that for smooth curves over a local field or surfaces over a finite field, replacing the Witt group by the total Witt group of all line bundle-valued quadratic forms recovers Merkurjev's th…
The Tate conjecture for divisors on varieties over number fields is equivalent to finiteness of $\ell$-primary torsion in the Brauer group. We show that this finiteness is actually uniform in one-dimensional families for varieties that satisfy the Tate conjecture for divisors -- e.g. abelian varieties and $K3$ surfaces.
Using the calculus of Steenrod operations in étale cohomology developed in [Feng17], we prove that the analogue of Tate's pairing on higher Brauer groups is alternating on 2-torsion. This improves upon a result of Jahn [Jahn15, Math. Annalen].
Cohomological invariants and Brauer groups of algebraic stacks in positive characteristic
引入特征p下代数栈的带模p^r系数上同调不变量理论，并据此完成了任意域上椭圆曲线栈的Brauer群与上同调不变量的计算。
Differential forms and Brauer classes in positive characteristic
本文研究正特征下微分形式导出的p挠Brauer类，关联其与超奇异K3曲面Brauer群的关系，分析其对Brauer-Manin障碍的贡献，并考察相关曲面与簇的Brauer-Manin集。
Let X --> P^2 be a p-cyclic cover branched over a smooth, connected curve C of degree divisible by p, defined over a separably closed field of prime-to-p characteristic. We show that all (unramified) p-torsion Brauer classes on X that are fixed by Aut(X/P^2) arise as pullbacks of certain Brauer classes on k(P^2) that are unramified away from C and a fixed line L. We completely characterize these Brauer classes on k(P^2) and relate the kernel of the pullback map to the Picard group of X. If p = 2, we give a second construction, which works over any base field of characteristic not 2, that uses …
Unramified Brauer groups of symmetric products and the Brauer-Manin obstructions
研究光滑、射影且几何积分的变种X，其几何Picard群为 torsion-free，建立X与对称积的Brauer群同构，探讨Brauer-Manin障碍对Hasse原理和弱逼近及理点障碍的关系。
Dichotomy for the Brauer-Manin obstruction in characteristic p
研究特征p全局函数域上光滑射影几何整变种的p-主Brauer-Manin障碍，证明当单幂Brauer群为零且Picard概形无p-挠时，仅有限个位可能相关；否则几乎所有位都可能相关。
Brauer group of varieties over local fields of finite characteristic
本文在正特征局部域上研究了代数簇的Brauer群，证明了Kato挠 filtration 与评估 filtration 的等价性，并扩展了相关结果到正特征情况。
We study Kummer varieties attached to 2-coverings of abelian varieties of arbitrary dimension. Over a number field we show that the subgroup of odd order elements of the Brauer group does not obstruct the Hasse principle. Sufficient conditions for the triviality of the Brauer group are given, which allow us to give an example of a Kummer K3 surface of geometric Picard rank 17 over the rationals with trivial Brauer group. We establish the non-emptyness of the Brauer--Manin set of everywhere locally soluble Kummer varieties attached to 2-coverings of products of hyperelliptic Jacobians with larg…
Soit K le corps des fractions d'une surface projective et lisse, géométriquement intègre, définie sur un corps fini F. Soit C/K une conique. Parimala et Suresh ont montré que le groupe de cohomologie non ramifiée en degré trois de K(C) à coefficients de torsion est nul. Dans cette note on étend leur résultat aux variétés de Severi-Brauer associées à une algèbre centrale simple dont d'indice l est premier et différent de char F.
We give bounds on the order of torsion in the Chow group of zero dimensional cycles for isotropic Grassmannians and Brauer-Severi flag varieties. To do this, we introduce tools to understand the behavior of torsion in Chow groups with coefficients under morphisms of proper varieties
We exhibit central simple algebras over the function field of a diagonal quartic surface over the complex numbers that represent the 2-torsion part of its Brauer group. We investigate whether the 2-primary part of the Brauer group of a diagonal quartic surface over a number field is algebraic and give sufficient conditions for this to be the case. In the last section we give an obstruction to weak approximation due to a transendental class on a specific diagonal quartic surface, an obstruction which cannot be explained by the algebraic Brauer group which in this case is just the constant algeb…
We provide an alternative proof that the Chow group of $1$-cycles on a Severi--Brauer variety associated to a biquaternion division algebra is torsion-free. There are three proofs of this result in the literature, all of which are due to Karpenko and rely on a clever use of $K$-theory. The proof that we give here, by contrast, is geometric and uses degenerations of quartic elliptic normal curves.
We show that for complex analytic K3 surfaces any torsion class in H^2(X,O_X^*) comes from an Azumaya algebra. In other words, the Brauer group equals the cohomological Brauer group. For algebraic surfaces, such results go back to Grothendieck. In our situation, we use twistor spaces to deform a given analytic K3 surface to suitable projective K3 surfaces, and then stable bundles and hyperholomorphy conditions to pass back and forth between the members of the twistor family.
A purity conjecture due to Grothendieck and Auslander--Goldman predicts that the Brauer group of a regular scheme does not change after removing a closed subscheme of codimension $\ge 2$. The combination of several works of Gabber settles the conjecture except for some cases that concern $p$-torsion Brauer classes in mixed characteristic $(0, p)$. We establish the remaining cases by using the tilting equivalence for perfectoid rings. To reduce to perfectoids, we control the change of the Brauer group of the punctured spectrum of a local ring when passing to a finite flat cover.
Brauer--Manin obstruction from failure of Kodaira vanishing in positive characteristic
研究正特征全局域上 Kodaira 消失定理失效的光滑射影曲面的 Brauer-Manin 集，特别应用于 Raynaud 曲面。
Let X be a smooth double cover of a geometrically ruled surface defined over a separably closed field of characteristic different from 2. The main result of this paper is a finite presentation of the 2-torsion in the Brauer group of X with generators given by central simple algebras over the function field of X and relations coming from the Néron-Severi group of X. In particular, the result gives a central simple algebra representative for the unique nontrivial Brauer class on any Enriques surface. An example demonstrating the applications to the study of rational points is given.
Over a normal base scheme, we prove the generalized Theorem of the Cube for 1-motives and that a torsion class of the group H^2_ét(M,G_m)$ of a 1-motive M, whose pull-back via the unit section is zero, comes from an Azumaya algebra. In particular, we deduce that over an algebraically closed field of characteristic zero, all classes of H^2_ét(M,G_m) come from Azumaya algebras.
We analyze the Brauer-Manin obstruction to rational points on the K3 surfaces over $\mathbb{Q}$ given by double covers of $\mathbb{P}^2$ ramified over a diagonal sextic. After finding an explicit set of generators for the geometric Picard group of such a surface, we find two types of infinite families of counterexamples to the Hasse principle explained by the algebraic Brauer-Manin obstruction. The first type of obstruction comes from a quaternion algebra, and the second type comes from a 3-torsion element of the Brauer group, which gives an affirmative answer to a question asked by Ieronymou …
For a prime $p$, we study subgroups of order p of the Brauer group Br(S) of a general complex polarized K3 surface of degree 2d, generalizing earlier work of van Geemen. These groups correspond to sublattices of index p of the transcendental lattice T_S of S; we classify these lattices up to isomorphism using Nikulin's discriminant form technique. We then study geometric realizations of p-torsion Brauer elements as Brauer-Severi varieties in a few cases via projective duality. We use one of these constructions for an arithmetic application, giving new kinds of counter-examples to weak approxim…
We compute the Brauer group of the moduli stack of hyperelliptic curves $\mathcal{H}_g$ over any field of characteristic zero. In positive characteristic, we compute the part of the Brauer group whose order is prime to the characteristic of the base field.
We study the semi-decomposable invariants of a split semisimple group and their extension to a split reductive group by using the torsion in the codimension $2$ Chow groups of a product of Severi-Brauer varieties. In particular, for any $n\geq 2$ we completely determine the degree $3$ invariants of a split semisimple group, the quotient of $(\operatorname{\mathbf{SL}}_{2})^{n}$ by its maximal central subgroup, as well as of the corresponding split reductive group. We also provide an example illustrating that a modification of our method can be applied to find the semi-decomposable invariants o…
Let $F$ be the function field of a smooth curve over the $p$-adic number field $\Q_p$. We show that for each prime-to-$p$ number $n$ the $n$-torsion subgroup $\H^2(F,μ_n)={}_n\Br(F)$ is generated by $\Z/n$-cyclic classes; in fact the $\Z/n$-length is equal to two. It follows that the Brauer dimension of $F$ is two (first proved in \cite{Sa97}), and any $F$-division algebra of period $n$ and index $n^2$ is decomposable.
For a K3 surface over a field of characteristic 2 which is finitely generated over its prime subfield, we prove that the cokernel of the natural map from the Brauer group of the base field to that of the K3 surface is finite modulo the 2-primary torsion subgroup. In characteristic different from 2, such results were previously proved by A. N. Skorobogatov and Y. G. Zarhin. We basically follow their methods with an extra care in the case of superspecial K3 surfaces using the recent results of W. Kim and K. Madapusi Pera on the Kuga-Satake construction and the Tate conjecture for K3 surfaces in …
The Brauer-Manin obstruction on algebraic stacks
本文研究代数堆上的Brauer-Manin集和配对，扩展了Colliot-Thélène和Sansuc的降理论，证明了局部由变分线性群商的堆的Brauer群的 torsion 性，并展示了Brauer-Manin障碍与其它上同调障碍的一致性。
Let X be a smooth variety over a number field k embedded as a degree d subvariety of $\mathbb{P}^n$ and suppose that X is a counterexample to the Hasse principle explained by the Brauer-Manin obstruction. We consider the question of whether the obstruction is given by the d-primary subgroup of the Brauer group, which would have both theoretic and algorithmic implications. We prove that this question has a positive answer in the case of torsors under abelian varieties, Kummer surfaces and (conditional on finiteness of Tate-Shafarevich groups) bielliptic surfaces. For Kummer varieties we show th…
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