Related papers: Differential forms and Brauer classes in positive …

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…

We consider the reduction of Brauer classes on surfaces over number fields, with a view toward applications to rationality and derived equivalence. We show that a Brauer class on a very general polarized K3 surface over a number field…

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…

We study the geometry of exceptional loci of birational contractions of hyper-K\"ahler fourfolds that are of K3$^{[2]}$-type. These loci are conic bundles over K3 surfaces and we determine their classes in the Brauer group. For this we use…

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.

We study obstructions to the existence of Riemannian metrics of positive scalar curvature on closed smooth manifolds arising from torsion classes in the integral homology of their fundamental groups. As an application, we construct new…

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…

We compute divisors class groups of singular surfaces. Most notably we produce an exact sequence that relates the Cartier divisors and almost Cartier divisors of a surface to the those of its normalization. This generalizes Hartshorne's…

We investigate obstruction classes of moduli spaces of sheaves on K3 surfaces. We extend previous results by Caldararu, explicitly determining the obstruction class and its order in the Brauer group. Our main theorem establishes a short…

Deligne 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…

We generalise the notion of the Tate-Shafarevich group of an elliptic K3 surface with a section to the Tate-Shafarevich group of a K3 surface endowed with a linear system. The construction, which uses Grothendieck's special Brauer group,…

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…

Let $X$ be a K3 surface defined over a number field $k$, with principal complex multiplication by a CM field $E$. We find explicit bounds, in terms of $k$ and $E$, on the size of the transcendental Brauer group…

In this paper we will use a particular non-commutative scheme to, among other things, study the ramification properties of the field of $p$-torsion points on an elliptic curve and its reduction properties. Also, we show that this…

On 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.

In this paper we study the existence of rational points for the family of K3 surfaces over $\mathbb{Q}$ given by $$w^2 = A_1x_1^6 + A_2x_2^6 + A_3x_3^6.$$ When the coefficients are ordered by height, we show that the Brauer group is almost…

We construct families of log K3 surfaces and study the arithmetic of their members. We use this to produce explicit surfaces with an order 5 Brauer-Manin obstruction to the integral Hasse principle.

We show that transcendental elements of the Brauer group of an algebraic surface can obstruct the Hasse principle. We construct a general K3 surface X of degree 2 over Q, together with a two-torsion Brauer class A that is unramified at…

We provide a relation between Brauer-Manin obstruction and descent obstruction for torsors over open varieties under a connected linear algebraic group or a group of multiplicative type is given. Such a relation is further refined for…

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…