Qixiao Ma
Let $X$ be a smooth proper curve defined over a field $k$. The representability of the relative Picard functor is obstructed by a class $α\in\mathrm{Br}(\mathrm{Pic}_{X/k})$. We show the associated division algebra on $\mathrm{Pic}^0_{X/k}$ has natural involutions. We show the class $α$ splits at some height one points in $\mathrm{Pic}_{X/k}$.
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We examine the noncommutative minimal model program for orders on arithmetic surfaces, or equivalently, arithmetic surfaces enriched by a Brauer class $β$. When $β$ has prime index $p>5$, we show the classical theory extends with analogues of existence of terminal resolutions, Castelnuovo contraction and Zariski factorisation. We also classify $β$-terminal surfaces and Castelnuovo contractions, and discover new unexpected behaviour.
We study the Brauer class rising from the obstruction to the existence of tautological line bundles on the Picard scheme of curves. We determine the period and index of the Brauer class in certain cases.
Splitting unramified Brauer classes by abelian torsors and the period-index problem
本文利用扭曲相对Picard簇，通过固定阿贝尔概型的挠子分裂射影簇上的Brauer类，并证明非分歧Brauer类的指标整除其周期的固定幂次。
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 prove new results on splitting Brauer classes by genus 1 curves, settling in particular the case of degree 7 classes over global fields. Though our method is cohomological in nature, and proceeds by considering the more difficult problem of splitting $μ_N$-gerbes, we use crucial input from the arithmetic of modular curves and explicit $N$-descent on elliptic curves.
We prove that the index of a Brauer class satisfies prime decomposition over a general base scheme. This contrasts with our previous result that there is no general prime decomposition of Azumaya algebras.
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。
This paper studies etale twists of derived categories of schemes and associative algebras. A general method, based on a new construction called the twisted Brauer space, is given for classifying etale twists, and a complete classification is carried out for genus 0 curves, quadrics, and noncommutative projective spaces. A partial classification is given for curves of higher genus. The techniques build upon my recent work with David Gepner on the Brauer groups of commutative ring spectra.
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.
In this paper we introduce a method to obtain algebraic information using arithmetic one in the study of tori and their principal homogeneous spaces. In particular, using some results of the authors with Tingyu Lee, we determine the unramified Brauer groups of some norm one tori, and their torsors.
On Brauer groups of known Enriques manifolds
研究已知恩里ques manifold的Brauer群，构建特殊Brauer-Severi变种，并从几何和代数角度研究其到超Kähler universal cover Brauer群的拉回映射。
We classify morphisms from proper varieties to Brauer-Severi varieties, which generalizes the classical correspondence between morphisms to projective space and globally generated invertible sheaves. As an application, we study del Pezzo surfaces of large degree with a view towards Brauer-Severi varieties, and recover classical results on rational points, the Hasse principle, and weak approximation.
We show that in locally-ringed connected topoi the primes dividing the period and index of a Brauer class coincide. The result applies in particular to Brauer classes on connected schemes, algebraic stacks, topological spaces and to the projective representation theory of profinite groups.
Brauer classes not split by genus one curves
证明存在域F上的Brauer类不能被任何F上的亏格一曲线分裂，回答了Clark和Saltman的问题。
We prove that the moduli spaces of framed bundles over a smooth projective curve are rational. We compute the Brauer group of these moduli spaces to be zero under some assumption on the stability parameter.
We describe a method to compute the Brauer-Manin obstruction for smooth cubic surfaces over $\bbQ$ such that $\Br(S)/\Br(\bbQ)$ is a 3-group. Our approach is to associate a Brauer class with every ordered triplet of Galois invariant pairs of Steiner trihedra. We show that all order three Brauer classes may be obtained in this way. To show the effect of the obstruction, we give explicit examples.
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).
Effective Brauer-Siegel on some curves in $Y(1)^n$
本文通过 Yves André 的 G 函数方法，在某些 $Y(1)^n$ 曲线上建立了二次虚数域类数的有效 Siegel 下界。
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 $S$ 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.
Splitting Brauer classes by genus one curves over number fields
该研究证明数域上维数≥2的Severi-Brauer簇含扭曲椭圆正规曲线，进而得出数域上每个Brauer类都被亏格1曲线分裂的结论，所用方法为纤维化方法，核心是证明相关Hilbert概型光滑紧化的有理点在其Brauer-Manin集内稠密。
Vanishing of Brauer classes on K3 surfaces under reduction
本文证明了数域上K3曲面的Brauer类在某些技术假设下存在无穷多个约化使其消失，回答了Frei--Hassett--Várilly-Alvarado的问题。
We prove some new relations between weak approximation and some rational equivalence relations (Brauer and R-equivalence) in algebraic groups over arithmetical fields. By using weak approximation and local - global approach, we compute completely the group of Brauer equivalence classes of connected linear algebraic groups over number fields, and also completely compute the group of R-equivalence classes of connected linear algebraic groups $G$, which either are defined over a totally imaginary number field, or contains no anisotropic almost simple factors of exceptional type $^{3,6}\D_4$, nor …
Brauer-Manin obstruction for Wehler K3 surfaces of Markoff type
本研究针对Markoff型Wehler K3曲面整数点的Brauer-Manin障碍展开，构造了违背整数Hasse原理的族并给出强逼近的反例，还研究了仿射曲面有理点的相关问题。
Let k be a field, X a smooth, projective k-variety. If X is geometrically rational, there is an injective map from the quotient of Brauer groups Br(X)/Br(k) into the first Galois cohomology group of the lattice given by the geometric Picard group. In this note, where the main attention is on smooth compactifications of homogeneous spaces of algebraic k-tori, we show how under some hypotheses the map is onto, and how one may in some special case exhibit concrete generators in Br(X). This is applied to the analysis of counterexamples to the local-global principle for norms in biquadratic extensi…
Invariants of Vanishing Brauer Classes
研究Picard秩为1的K3曲面特殊化到秩为2时定义的阶为2消失Brauer类，给出其B-场不变量，并应用于带平面的三次四重定义的K3双重平面，确定其Picard格及与Clifford类的关系，从而显式确定相关K3曲面。
We reinterpret the residue map for the Brauer group of a smooth variety using a root stack construction and Weil restriction for algebraic stacks, and apply the result to find a geometric representative of for the residue of a Brauer class associated to a conic bundle, or more generally a Brauer-Severi bundle.
We show that Brauer classes of a locally solvable degree 4 del Pezzo surface X are vertical for some projection away from a plane f: X ---> P^1, i.e., that every Brauer class is obtained by pullback from an element of Br k(P^1). As a consequence, we prove that a Brauer class does not obstruct the existence of a rational point if and only if there exists a fiber of f that is locally solvable. The proof is constructive and gives a simple and practical algorithm, distinct from that in [BBFL07], for computing all classes in the Brauer group of X (modulo constant algebras).
On the Brauer class of Modular Endomorphism Algebras
本文研究非CM形式所附带的动机的端射影代数的Brauer类，并揭示其分枝性与归一化斜率的关系。
Some remarks on Brauer Classes of K3-type
研究了复射影 K3 曲面 Brauer 群中素数阶元素定义的模空间有限映射，计算了其次数，并揭示了 Picard 数为 2 时 Picard 格的决定关系。
We show that if over some number field there exists a certain diagonal plane cubic curve that is locally solvable everywhere, but that does not have points over any cubic galois extension of the number field, then the algebraic part of the Brauer-Manin obstruction is not the only obstruction to the Hasse principle for K3 surfaces.
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…
For an affine double plane defined by an equation of the form z^2 = f, we study the divisor class group and the Brauer group. Two cases are considered. In the first case, f is a product of n linear forms in k[x,y] and X is birational to a ruled surface P^1 x C, where C is rational if n is odd and hyperelliptic if n is even. In the second case, f is the equation of an affine hyperelliptic curve. On the open set where the cover is unramified, we compute the groups of divisor classes, the Brauer groups, the relative Brauer group, as well as all of the terms in he exact sequence of Chase, Harrison…
Let $\text{M}_C( 2, \mathcal{O}_C) \cong \mathbb{P}^3$ denote the coarse moduli space of semistable vector bundles of rank $2$ with trivial determinant over a smooth projective curve $C$ of genus $2$ over $\mathbb{C}$. Let $β_C$ denote the natural Brauer class over the stable locus. We prove that if $f^*( β_{C'}) = β_C$ for some birational map $f$ from $\text{M}_C( 2, \mathcal{O}_C)$ to $\text{M}_{C'}( 2, \mathcal{O}_{C'})$, then the Jacobians of $C$ and of $C'$ are isomorphic as abelian varieties. If moreover these Jacobians do not admit real multiplication, then the curves $C$ and $C'$ are i…
The Brauer group of $BG$ and gerbe structures of moduli spaces
本文研究了具有阶$N$自同构的曲线的$μ_N$-gerbe结构，并确定了其Brauer类，关键在于计算$BG$的Brauer群。
We classify absolutely split vector bundles on proper $k$-schemes. More precise, we prove that the closed points of the Picard scheme are in one-to-one correspondence with indecomposable absolutely split vector bundles. Furthermore, we apply the obtained results to study the geometry of (generalized) Brauer--Severi varieties.
Class field theory, Hasse principles and Picard-Brauer duality for two-dimensional local rings
本文基于具有完美剩余域的二维局部环的算术对偶，通过分析算术上同调的ind-pro代数群结构并证明其有限性性质，推导出了类域论、Hasse原理及除子类群与Brauer群的对偶。
In this paper we investigate the $\mathbb{Q}$-rational points of a class of simply connected Calabi-Yau threefolds, which were originally studied by Hosono and Takagi in the context of mirror symmetry. These varieties are defined as a linear section of a double quintic symmetroid; their points correspond to rulings on quadric hypersurfaces. They come equipped with a natural $2$-torsion Brauer class. Our main result shows that under certain conditions, this Brauer class gives rise to a transcendental Brauer-Manin obstruction to weak approximation. Hosono and Takagi showed that over $\mathbb{C}$…
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 $\mathrm{Br}(E)/\mathrm{Br}(k)$.
Let $\{P_i\}_{1 \leq i \leq r}$ and $\{Q_i\}_{1 \leq i \leq r}$ be two collections of Brauer Severi surfaces (resp. conics) over a field $k$. We show that the subgroup generated by the $P_i's$ in $Br(k)$ is the same as the subgroup generated by the $Q_i's$ \iff $ΠP_i $ is birational to $ΠQ_i$. Moreover in this case $ΠP_i$ and $ΠQ_i$ represent the same class in $M(k)$, the Grothendieck ring of $k$-varieties. The converse holds if $char(k)=0$. Some of the above implications also hold over a general noetherian base scheme.
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。
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…
Brauer groups of conic bundles over elliptic curves
本文研究数域上椭圆曲线的正则圆锥丛的Brauer群，在奇异纤维位于2-可除点且分歧域同构的条件下给出显式计算，并应用于一类Châtelet型曲面，进而分析实位点产生的Brauer-Manin障碍对弱逼近的影响。
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.
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