Jan de Gier, Bernard Nienhuis
We observe that the degree of the commuting variety and other related varieties occur as coefficients in the leading eigenvector of an integrable loop model based on the Brauer algebra.
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We introduce the_Brauer loop scheme_ E := {M in M_N(C) : M\cp M = 0}, where \cp is a certain degeneration of the ordinary matrix product. Its components of top dimension, floor(N^2/2), correspond to involutions πin S_N having one or no fixed points. In the case N even, this scheme contains the upper-upper scheme from [Knutson '04] as a union of (N/2)! of its components. One of those is a degeneration of the_commuting variety_ of pairs of commuting matrices. The_Brauer loop model_ is a quantum integrable stochastic process introduced in [de Gier--Nienhuis '04], and some of the entries of its Pe…
Let A and B be two central simple algebras of a prime degree n over a field F generating the same subgroup in the Brauer group. We show that the Chow motive of a Severi-Brauer variety SB(A) is a direct summand of the motive of a generalized Severi-Brauer variety SB_d(B) if and only if [A]=d[B] or [A]=-d[B] in the Brauer group. The proof uses methods of Schubert calculus and combinatorial properties of Young tableaux, e.g., Robinson-Schensted correspondence.
We prove the existence of Ulrich bundles on any Brauer--Severi variety. In some cases, the minimal possible rank of the obtained Ulrich bundles equals the period of the Brauer--Severi variety. Moreover, we find a formula for the rank of an Ulrich bundle involving the period of the considered Brauer--Severi variety $X$, at least if $\mathrm{dim}(X)=p-1$ for an odd prime $p$. This formula implies that the rank of any Ulrich bundle on such a Brauer--Severi variety $X$ must be a multiple of the period.
We study a new object that can be attached to an abelian variety or a complex torus: the invariant Brauer group, as recently defined by Yang Cao. Over the field of complex numbers this is an elementary abelian 2-group with an explicit upper bound on the rank. We exhibit many cases in which the invariant Brauer group is zero, and construct complex abelian varieties in every dimension starting with 2, both simple and non-simple, with invariant Brauer group of order 2. We also address the situation in finite characteristic and over non-closed fields.
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.
Let X be the variety obtained by the Weil transfer with respect to a quadratic separable field extension of a generalized Severi-Brauer variety. We study (and, in some cases, determine) the canonical dimension, incompressibility, and motivic indecomposability of X. We determine the canonical 2-dimension of X (in the general case).
J. Stix proved that a curve of positive genus over $\mathbb{Q}$ which maps to a non-trivial Brauer-Severi variety satisfies the section conjecture. We prove that, if $X$ is a curve of positive genus over a number field $k$ and the Weil restriction $R_{k/\mathbb{Q}}X$ admits a rational map to a non-trivial Brauer-Severi variety, then $X$ satisfies the section conjecture. As a consequence, if $X$ maps to a Brauer-Severi variety $P$ such that the corestriction $\operatorname{cor}_{k/\mathbb{Q}}([P])\in\operatorname{Br}(\mathbb{Q})$ is non-trivial, then $X$ satisfies the section conjecture.
The purpose of this article is to show how one might compute the étale cohomology groups $H^p_{\acute{e}t}(X,G_m)$ in degrees $p=0$, $1$ and $2$ of a toric variety $X$ with coefficients in the sheaf of units. The method is to reduce the computation down to the problem of diagonalizing a matrix with integral coefficients. The procedure outlined in this article has been fully implemented by the author as a program written in the ``C'' programming language.
We study homotopy rational points of Brauer-Severi varieties over fields of characteristic zero. We are particularly interested if a Brauer-Severi variety admitting a homotopy rational point splits. The analogue statement turns out to be true for open subvarieties of twisted hyperplane arrangements. In the complete case, the obstruction turns out to be the pullback of the Chern class of a generator of the Picard group along the homotopy rational point. Moreover, we will show that every Brauer-Severi variety over a $p$-adic field with $p$-primary period admits a homotopy rational point.
Added lemma provided by Michel Brion. Other (minor) changes. Submitted version. Let k be any field, let X' be a projective and geometrically integral k-scheme and let Y' be a finite closed subscheme of X'. If f: Y'-> Y is a schematically dominant morphism between finite k-schemes and X is obtained by pinching X' along Y' via f, we describe the kernel (and, in certain cases, the cokernel) of the induced pullback map Br_{1}X -> Br_{1}X' between the corresponding algebraic (cohomological) Brauer groups of X and X'_solely_ in terms of the Brauer groups of the residue fields of Y and Y' and the Ami…
For a quasi-projective smooth geometrically integral variety over a number field $k$, we prove that the iterated descent obstruction is equivalent to the descent obstruction. This generalizes a result of Skorobogatov, and this answers an open question of Poonen. The key idea is the notion of invariant Brauer subgroup and the notion of invariant étale Brauer-Manin obstruction for a $k$-variety equipped with an action of a connected linear algebraic group.
We classify products of symmetric powers of a Severi-Brauer variety, up-to stable birational equivalence. The description also includes Grassmannians and moduli spaces of genus 0 stable maps.
Toric varieties are a special class of rational varieties defined by equations of the form {\it monomial = monomial}. For a good brief survey of the history and role of toric varieties see [10]. Any toric variety $X$ contains a cover by affine open sets described in terms of arrangements (called fans) of convex bodies in $\Bbb R^r$. The coordinate rings of each of these affine open sets is a graded ring generated over the ground field by monomials. As a consequence, toric varieties provide a good context in which cohomology can be calculated. The purpose of this article is to describe the seco…
The étale Brauer\textendash Manin obstruction is equivalent to each other among Weil restrictions of an arbitrarily given quasi-projective algebraic variety defined over a number field.
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。
For $X$ a smooth projective variety over a field $k$, we consider the problem of Galois descent for higher Brauer groups. More precisely, we extend a finiteness result of Colliot-Thélène and Skorobogatov to higher Brauer groups.
Refined index obstructions for Brauer classes on an abelian variety
本文为光滑射影复簇上的拓扑平凡 Brauer 类构造了精细指数障碍，推广了 de Jong 和 Perry 的最新结果，并证明了这些障碍比之前的更严格，从而给出了更多整 Hodge 猜想的反例，同时重点展示了在复阿贝尔簇上的算法实现。
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.
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.
Let $X$ be a projective and smooth variety over a field $k$. The goal of this paper is to prove that the cokernel of the canonical map $Br(X)\to Br(X_{k^s})^{G_k}$ has a finite exponent. Both groups are natural invariants arising from consideration of the Tate conjecture of divisors over $X$.
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.
Let $X$ be a smooth projective variety with a fibration into varieties that either satisfy a condition on representability of zero-cycles or that are torsors under an abelian variety. We study the classes in the Brauer group that never obstruct the Hasse principle for $X$. We prove that if the generic fiber has a zero-cycle of degree $d$ over the generic point, then the Brauer classes whose orders are prime to $d$ do not play a role in the Brauer--Manin obstruction. As a result we show that the odd torsion Brauer classes never obstruct the Hasse principle for del Pezzo surfaces of degree 2, ce…
Let p be a positive prime number and X be a Severi-Brauer variety of a central division algebra D of degree p^n, with n>0. We describe all shifts of the motive of X in the complete motivic decomposition of a variety Y, which splits over the function field of X and satisfies the nilpotence principle. In particular, we prove the motivic decomposability of generalized Severi-Brauer varieties X(p^m,D) of right ideals in D of reduced dimension p^m, m=0,1,...,n-1, except the cases p=2, m=1 and m=0 (for any prime p), where motivic indecomposability was proven by Nikita Karpenko.
Brauer-Manin obstructions requiring arbitrarily many Brauer classes
本文研究定义在整体域上的射影簇，证明捕获 Brauer-Manin 障碍的 Brauer 群有限子群可能需要任意多个生成元。
For a smooth and projective variety X over a field k of characteristic zero we prove the finiteness of the cokernel of the natural map from the Brauer group of X to the Galois-invariant subgroup of the Brauer group of the same variety over an algebraic closure of k. Under further conditions on k, e.g. over number fields, we give estimates for the order of this cokernel. ---- Soit X une variété projective et lisse sur un corps k de caractéristique zéro. Le groupe de Brauer de X s'envoie dans les invariants, sous le groupe de Galois absolu de k, du groupe de Brauer de la même variété considérée …
We produce a partial compactification of the variety given by P(t)=N_{K/k}(\mathbf z) whose Brauer group coincides with the unramified Brauer group, where K is an étale k-algebra and P(t)\in k[t] is a nonconstant polynomial. Then we obtain a systematic method to compute the unramified Brauer group for all such varieties.
Since Poonen's construction of a variety $X$ defined over a number field $k$ for which $X(k)$ is empty and the étale Brauer--Manin set $X(\mathbf{A}_k)^\text{Br,et}$ is not, several other examples of smooth, projective varieties have been found for which the étale Brauer--Manin obstruction does not explain the failure of the Hasse principle. All known examples are constructed using "Poonen's trick", i.e. they have the distinctive feature of being fibrations over a higher genus curve; in particular, their Albanese variety is non-trivial. In this paper, we construct examples for which the Albane…
Finite subgroups of automorphism groups of Severi--Brauer varieties of prime degree
本文研究了素数维度的非平凡塞弗里-布拉泽变元的自同构群的有限子群分类，并构造了示例，证明了这些变元在特征不等于素数的域上不G-理性刚性。
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。
The commuting variety of a reductive Lie algebra ${\goth g}$ is the underlying variety of a well defined subscheme of $\gg g{}$. In this note, it is proved that this scheme is normal. In particular, its ideal of definition is a prime ideal.
For a reductive Lie algbera over an algbraically closed field of charasteristic zero,we consider a borel subgroup $B$ of its adjoint group, a Cartan subalgebra contained inthe Lie algebra of $B$ and the closure $X$ of its orbit under $B$ in the Grassmannian.The variety $X$ plays an important role in the study of the commuting variety. In thisnote, we prove that $X$ is Gorenstein with rational singularities.
Let G be a semisimple affine algebraic group of inner type over a field F. We write C for the class of all finite direct products of projective G-homogeneous F-varieties. We determine the structure of the Chow motives with coefficients in a finite field of the varieties in C. More precisely, it is known that the motive of any variety in C decomposes (in a unique way) into a sum of indecomposable motives, and we describe the indecomposable summands which appear in the decompositions. In the case where G is the group of automorphisms of a given central simple F-algebra A, for any variety in the …
We prove that every Brauer class over a field splits over a torsor under an abelian variety. If the index of the class is not congruent to 2 modulo 4, we show that the Albanese variety of any smooth curve of positive genus that splits the class also splits the class, and there exist many such curves splitting the class. We show that this can be false when the index is congruent to 2 modulo 4, but adding a single genus 1 factor to the Albanese suffices to split the class.
The aim of this paper is to describe the irregular locus of the commuting variety of a reductive symmetric Lie algebra. More precisely, we want to enlighten a remark of Popov. In [Po], the irregular locus of the commuting variety of any reductive Lie algebra is described and its codimension is computed. This provides a bound for the codimension of the singular locus of this commuting variety. [Po, Remark 1.13] suggests that the arguments and methods of [Po] are suitable for obtaining analogous results in the symmetric setting. We show that some difficulties arise in this case and we obtain som…
For a smooth proper variety over a $p$-adic field, the Brauer group and abelian fundamental group are related to the higher Chow groups by the Brauer-Manin pairing and the class field theory. We generalize this relation to smooth (possibly non-proper) varieties, using the motivic homology and the tame version of Wiesend's ideal class group. Several examples are discussed.
Let $k$ be a field that is finitely generated over the field of rational numbers and $Br(k)$ the Brauer group of $k$. Let $X$ be an absolutely irreducible smooth projective variety over $k$, let $Br(X)$ be the cohomological Brauer-Grothendieck group of $X$ and $Br_0(X)$ the image of $Br(k)$ in $Br(X)$. We write $Br_1(X)$ for the subgroup of elements in $Br(X)$ that become trivial after replacing $k$ by its algebraic closure. We prove that $Br(X)/Br_0(X)$ is finite if $X$ is a $K3$ surface. When $X$ is (a principal homogeneous space of) an abelian variety over $k$ then we prove that $Br(X)/Br_1…
The_commuting variety_ is the pairs of NxN matrices (X,Y) such that XY = YX. We introduce the_diagonal commutator scheme_, {(X,Y) : XY-YX is diagonal}, which we prove to be a reduced complete intersection, one component of which is the commuting variety. (We conjecture there to be only one other component.) The diagonal commutator scheme has a flat degeneration to the scheme {(X,Y) : XY lower triangular, YX upper triangular}, which is again a reduced complete intersection, this time with n! components (one for each permutation). The degrees of these components give interesting invariants of pe…
Let K be a global field, let S be a finite set of primes of K containing the archimedean primes and let A be an abelian variety over K. We generalize the duality theorem established in our paper "On Neron class groups of abelian varieties" by removing the hypothesis in [op.cit.] that the Tate-Shafarevich group of A is finite. We also derive an exact sequence that relates the indicated group associated to the Jacobian variety of a proper, smooth and geometrically connected curve X over K to a certain finite subquotient of the Brauer group of X. The sequence alluded to above may be regarded as a…
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.
We study tropical commuting matrices from two viewpoints: linear algebra and algebraic geometry. In classical linear algebra, there exist various criteria to test whether two square matrices commute. We ask for similar criteria in the realm of tropical linear algebra, giving conditions for two tropical matrices that are polytropes to commute. From the algebro-geometric perspective, we explicitly compute the tropicalization of the classical variety of commuting matrices in dimension 2 and 3.
Pullback Method with Applications to Severi--Brauer Fibrations
本文提出一种拉回方法，用于构造具有布雷尔-曼因障碍物的代数簇，并展示存在索引为一的塞维-布拉泽纤维化违反哈斯原理。
We study the homogeneous irreducible Severi-Brauer varieties over an Abelian variety $A$. Such objects were classified by Brion, \cite{Bri}. Here we interpret that result within the context of cubical structures and biextensions for certain $\G_m$-torsors over finite subgroups of $A$. Our results can be seen as an instance of the theory developed by Breen, \cite{Breen:1983}, and Moret-Bailly, \cite{Moret-Bailly}.
The commuting variety of matrices over a given field is a well-studied object in linear algebra and algebraic geometry. As a set, it consists of all pairs of square matrices with entries in that field that commute with one another. In this paper we generalise the commuting variety by using the commuting distance of matrices. We show that over an algebraically closed field, each of our sets does indeed form a variety. We compute the dimension of the distance-$2$ commuting variety and characterize its irreducible components. We also work over other fields, showing that the distance-$2$ commuting…
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 study certain "σ-commuting varieties" associated with a pair of commuting involutions of a semisimple Lie algebra $\g$. The usual commuting variety of $\g$ and commuting varieties related to one involution are particular cases of our construction. We develop a general theory of σ-commuting varieties and point out some cases, when they have especially good properties. We show that, for a special choice of commuting involutions, the σ-commuting variety is isomorphic to the commuting variety of a simple Jordan algebra. As a by-product of our theory, we show that if $J$ is the Jordan algebra of…
We consider the set of forms of a toric variety over an arbitrary field: those varieties which become isomorphic to a toric variety after base field extension. In contrast to most previous work, we also consider arbitrary isomorphisms rather than just those that respect a torus action. We define an injective map from the set of forms of a toric variety to a non-abelian second cohomology set, which generalizes the usual Brauer class of a Severi-Brauer variety. Additionally, we define a map from the set of forms of a toric variety to the set of forms of a separable algebra along similar lines to…
We prove that for a large class of subvarieties of abelian varieties over global function fields, the Brauer-Manin condition on adelic points cuts out exactly the rational points. This result is obtained from more general results concerning the intersection of the adelic points of a subvariety with the adelic closure of the group of rational points of the abelian variety.
Let $X$ be a smooth projective variety over a finitely generated field $K$ of characteristic $p>0$. We proved that the finiteness of the $\ell$-primary part of $\mathrm{Br}(X_{K^s})^{G_K}$ for a single prime $\ell\neq p$ will imply the finiteness of the prime-to-$p$ part of $\mathrm{Br}(X_{K^s})^{G_K}$, generalizing a theorem of Tate and Lichtenbaum for varieties over finite fields. For an abelian variety $A$ over $K$, we proved a similar result for the Tate-Shafarevich group of $A$, generalizing a theorem of Schneider for abelian varieties over global function fields.
We study the anti-commuting variety which consists of pairs of anti-commuting $n\times n$ matrices. We provide an explicit description of its irreducible components and their dimensions. The GIT quotient of the anti-commuting variety with respect to the conjugation action of $GL_n$ is shown to be of pure dimension $n$. We also show the semi-nilpotent anti-commuting variety (in which one matrix is required to be nilpotent) is of pure dimension $n^2$ and describe its irreducible components.
The nilpotent cone of a reductive Lie algebra has a desingularization given by thecotangent bundle of the flag variety. Analogously, the nullcone of a cartesianpower of the algebra has a desingularization given by a vector bundle over theflag variety. As for the nullcone, the subvariety of elements whose componentsare in a same Borel subalgebra, has a desingularization given by a vector bundle overthe flag variety. In this note, some properties of these varieties are given. Forthe study of the commuting variety, the analogous variety to the flag variety isthe closure in the Grassmannian of the…
Projective dimension and commuting variety of a reductive Lie algebra
本文研究约化李代数交换簇，证明其对应子概形正规且Cohen-Macaulay（定义理想为素理想），并指出该结论源于单李代数的性质(P)（某些上同调复形正合）。
It is known that the variety of pairs of n x n commuting upper triangular matrices isn't a complete intersection for infinitely many values of n; we show that there exists m such that this happens if and only if n > m. We also show that m < 18 and that it could be found by determining the dimension of the variety of pairs of commuting strictly upper triangular matrices. Then we define a natural map from the variety of pairs of commuting n x n matrices onto a subvariety defined by linear equations of the grassmannian of subspaces of codimension 2 of a vector space of dimension n x n.
Let g be a complex reductive Lie algebra with Cartan algebra h. Hotta and Kashiwara defined a holonomic D-module M, on g x h, called Harish-Chandra module. We give an explicit description of gr(M), the associated graded module with respect to a canonical Hodge filtration on M. The description involves the isospectral commuting variety, a subvariety X of g x g x h x h which is a finite extension of the variety of pairs of commuting elements of g. Our main result establishes an isomorphism of gr(M) with the structure sheaf of the normalization of X. It follows, thanks to the theory of polarized …
Computing the equivariant Brauer group
该研究针对带有限阿贝尔群G正则作用的光滑射影有理簇，借助伽罗瓦上同调留数与固定轨迹几何，实现了二维、三维商叠[X/G]的Brauer群有效计算，并完成所有G-极小del Pezzo曲面对应Brauer群的计算。
Let K be a field. Let A be a central simple algebra over K and let X be the associated Brauer-Severi variety over K. It has recently been asked if there exists a genus 1 curve C over K such that K(C) splits A. In other words, is there a genus one curve C over K with a morphism to X? In this short note, we explicitly construct such a curve in the case where X has dimension at most 4 (equivalently, when A has degree at most 5).
Let g be a semisimple Lie algebra of finite dimension. The nullcone N of g is the set of (x,y) in g x g such that x and y are nilpotents and are in the same Borel sualgebra. The main result of this paper is that N is a closed and irreducible subvariety of g x g, its normalisation has rational singularities and its normalization morphism is bijective.
The commuting variety of $\mathfrak{pgl}_n$
本文研究了$\mathfrak{pgl}_n$的交换簇的结构，证明了在特定条件下其不可约分量的维数及相关代数结构的性质。
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。
In this paper we show that the derived category of Brauer-Severi curves satisfies the Jordan-Hölder property and cannot have quasi-phantoms, phantoms or universal phantoms. In this way we obtain that quasi-phantoms, phantoms or universal phantoms cannot exist in the derived category of smooth projective curves over a field $k$. Moreover, we show that a $n$-dimensional Brauer-Severi variety is completely characterized by the existence of a full weak exceptional collection consisting of pure vector bundles of length $n+1$, at least in characteristic zero. We conjecture that Brauer-Severi varieti…
Varieties with prescribed finite unramified Brauer groups and subgroups precisely obstructing the Hasse principle
针对数域上簇的Hasse原理阻碍问题，证明对任意非零有限阿贝尔群对$B_0\subset B$，存在射影线上的范丛簇$X$，其非分歧Brauer群同构于$B$且$B_0$是$B$中阻碍Hasse原理的最小子群。
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