Brauer Class Research Articles

K3 surfaces associated to a cubic fourfold

Let X⊂P5 be a smooth cubic fourfold. A well known conjecture asserts that X is rational if and only if there a Hodge theoretically associated K3 surface S. The surface S can be associated to X in two other different ways. If there is an equivalence of categories AX≃Db(S,α) where AX is the Kuznetsov component of Db(X) and α is a Brauer class, or if there is an isomorphism between the transcendental motive t(X) and the (twisted ) transcendental motive of a K3 surface S. In this note we consider families of cubic fourfolds with a finite group of automorphisms and describe the cases where there is an associated K3 surface in one of the above senses.

Kummer–Witt–Jackson algebras

This paper is concerned with the construction of a small, but non-trivial, example of a polynomial identity algebra, which we call the Jackson algebra, that will be used in sequels to this paper to study non-commutative arithmetic geometry. In this paper this algebra is studied from a ring-theoretic and geometric viewpoint. Among other things it turns out that this algebra is a “non-commutative family” of central simple algebras and thus parameterizes Brauer classes over extensions of the base.

Brauer–Manin obstructions requiring arbitrarily many Brauer classes

AbstractOn a projective variety defined over a global field, any Brauer–Manin obstruction to the existence of rational points is captured by a finite subgroup of the Brauer group. We show that this subgroup can require arbitrarily many generators.

Symbol length in positive characteristic

We show that any central simple algebra of exponent p in prime characteristic p that is split by a p-extension of degree pn is Brauer equivalent to a tensor product of 2⋅pn−1−1 cyclic algebras of degree p. If p=2 and n⩾3, we improve this result by showing that such an algebra is Brauer equivalent to a tensor product of 5⋅2n−3−1 quaternion algebras. Furthermore, we provide new proofs for some bounds on the minimum number of cyclic algebras of degree p that is needed to represent Brauer classes of central simple algebras of exponent p in prime characteristic p, which have previously been obtained by different methods.

Pinching Azumaya algebras

We show, that for a morphism of schemes from X to Y, that is a finite modification in finitely many closed points, a cohomological Brauer class on Y is represented by an Azumaya algebra if its pullback to X is represented by an Azumaya algebra. Part of the proof uses an extension of a result by Ferrand, on pinching of finite locally free sheaves, to Azumaya algebras.

ENRIQUES INVOLUTIONS AND BRAUER CLASSES

AbstractWe prove that every element of order 2 in the Brauer group of a complex Kummer surface X descends to an Enriques quotient of X. In generic cases, this gives a bijection between the set ${\mathcal Enr}(X)$ of Enriques quotients of X up to isomorphism and the set of Brauer classes of X of order 2. For some K3 surfaces of Picard rank $20,$ we prove that the fibers of ${\mathcal Enr}(X)\to \mathrm {{Br}}(X)[2]$ above the nonzero points have the same cardinality.

Reduction of Brauer classes on K3 surfaces, rationality and derived equivalence

Abstract Given a smooth projective variety over a number field and an element of its Brauer group, we consider the specialization of the Brauer class at a place of good reduction for the variety and the class. We are interested in the case of K3 surfaces. We show that a Brauer class on a very general polarized K3 surface over a number field becomes trivial after specialization at a set of places of positive natural density. We deduce that there exist cubic fourfolds over number fields that are conjecturally irrational, with rational reduction at a positive proportion of places. We also deduce that there are twisted derived equivalent K3 surfaces which become derived equivalent after reduction at a positive proportion of places.

Experiments on the Brauer map in high codimension

Using twisted and formal-local methods, we prove that every separated algebraic space which is the (open/flat) pushout of affine schemes has enough Azumaya algebras. As a corollary we show that, under mild hypothesis, every cohomological Brauer class is representable by an Azumaya algebra away from a closed subset of codimension $\geq 3$, generalizing an early result of Grothendieck. Next, we show that $\mathrm{Br}(X)=\mathrm{Br}'(X)$ when $X$ is an algebraic space obtained from a quasi-projective scheme by contracting a curve. This is the first method of descending an Azumaya algebra along a Chow cover which works in all dimensions. As a corollary, we prove that cohomological Brauer classes are geometric on arbitrary separated surfaces. In higher dimensions, we obtain the first examples of non-quasi-projective algebraic spaces with $\mathrm{Br}(X)=\mathrm{Br}'(X)$.

Specializing Brauer classes in Picard schemes

Let X be a smooth projective curve defined over a field k . The existence of tautological line bundles on X × Pic X / k is obstructed by a Brauer class α ∈ Br ( Pic X / k ) . We show that α splits at the generic points of various naturally defined loci in Pic X / k —the theta divisor and the generalized theta divisors associated with degree 2 g − 2 , semi-stable rank 2 vector bundles on X . We study a rank 2 analogue of Franchetta's theorem, and give explicit constructions of degree 2 g − 2 , semi-stable rank 2 vector bundles on the generic genus g curve.

A transcendental Brauer–Manin obstruction to weak approximation on a Calabi–Yau threefold

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 $\mathbb{C}$ each of these Calabi-Yau threefolds $Y$ is derived equivalent to a Reye congruence Calabi-Yau threefold $X$. We show that these derived equivalences may also be constructed over $\mathbb{Q}$, and we give sufficient conditions for $X$ to not satisfy weak approximation. In the appendix, N. Addington exhibits the Brauer groups of each class of Calabi--Yau variety over $\mathbb{C}$.

Lifts of Twisted K3 Surfaces to Characteristic 0

Abstract Deligne [9] showed that every K3 surface over an algebraically closed field of positive characteristic admits a lift to characteristic 0. We show the same is true for a twisted K3 surface. To do this, we study the versal deformation spaces of twisted K3 surfaces, which are particularly interesting when the characteristic divides the order of the Brauer class. We also give an algebraic construction of certain moduli spaces of twisted K3 surfaces over ${\operatorname {Spec}}\ \textbf {Z}$ and apply our deformation theory to study their geometry. As an application of our results, we show that every derived equivalence between twisted K3 surfaces in positive characteristic is orientation preserving.

Splitting Brauer classes using the universal Albanese

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.

Addition of Brauer classes via Picard schemes

We give an explicit way to construct a Brauer-Severi variety corresponding to the sum of the two Brauer classes that split by a common Galois extension. We apply the result to explicitly calculate the components of Picard schemes for stable curves with geometrically rational components.

Brauer class over the Picard scheme of totally degenerate stable curves

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 that Brauer class in certain cases.

Embedding of the derived Brauer group into the secondary $K$-theory ring

In this note, making use of the recent theory of noncommutative motives, we prove that the canonical map from the derived Brauer group to the secondary Grothendieck ring has the following injectivity properties: in the case of a regular integral quasi-compact quasi-separated scheme, it is injective; in the case of an integral normal Noetherian scheme with a single isolated singularity, it distinguishes any two derived Brauer classes whose difference is of infinite order. As an application, we show that the aforementioned canonical map is injective in the case of affine cones over smooth projective plane curves of degree \geq 4 as well as in the case of Mumford’s (famous) singular surface.

The topological period-index problem over $8$-complexes, II

We complete the study of the topological period-index problem over 8 dimensional finite CW complexes started in a preceding paper. More precisely, we determine the sharp upper bound of the index of a topological Brauer class $\alpha\in H^3(X;\mathbb{Z})$, where $X$ is of the homotopy type of an 8 dimensional finite CW complex and the period of $\alpha$ is divisible by 4.

On the non-neutral component of outer forms of the orthogonal group

Let (A,σ) be a central simple algebra with an orthogonal involution. It is well-known that O(A,σ) contains elements of reduced norm −1 if and only if the Brauer class of A is trivial. We generalize this statement to Azumaya algebras with orthogonal involution over semilocal rings, and show that the “if” part fails if one allows the base ring to be arbitrary.

Essential Dimension, Symbol Length and -rank

AbstractWe prove that the essential dimension of central simple algebras of degree $p^{\ell m}$ and exponent $p^{m}$ over fields $F$ containing a base-field $k$ of characteristic $p$ is at least $\ell +1$ when $k$ is perfect. We do this by observing that the $p$-rank of $F$ bounds the symbol length in $\text{Br}_{p^{m}}(F)$ and that there exist indecomposable $p$-algebras of degree $p^{\ell m}$ and exponent $p^{m}$. We also prove that the symbol length of the Kato-Milne cohomology group $\text{H}_{p^{m}}^{n+1}(F)$ is bounded from above by $\binom{r}{n}$ where $r$ is the $p$-rank of the field, and provide upper and lower bounds for the essential dimension of Brauer classes of a given symbol length.

Odd order obstructions to the Hasse principle on general K3 surfaces

We show that odd order transcendental elements of the Brauer group of a K3 surface can obstruct the Hasse principle. We exhibit a general K3 surface $Y$ of degree 2 over $\mathbb{Q}$ together with a three torsion Brauer class $\alpha$ that is unramified at all primes except for 3, but ramifies at all 3-adic points of $Y$. Motivated by Hodge theory, the pair $(Y, \alpha)$ is constructed from a cubic fourfold $X$ of discriminant 18 birational to a fibration into sextic del Pezzo surfaces over the projective plane. Notably, our construction does not rely on the presence of a central simple algebra representative for $\alpha$. Instead, we prove that a sufficient condition for such a Brauer class to obstruct the Hasse principle is insolubility of the fourfold $X$ (and hence the fibers) over $\mathbb{Q}_3$ and local solubility at all other primes.

The topological period–index problem over 8‐complexes, I

We study the Postnikov tower of the classifying space of a compact Lie group P ( n , m n ) , which gives obstructions to lifting a topological Brauer class of period n to a P U m n -torsor, where the base space is a CW complex of dimension 8 . Combined with the study of a twisted version of Atiyah–Hirzebruch spectral sequence, this solves the topological period–index problem for CW complexes of dimension 8.