Related papers: Brauer groups on K3 surfaces and arithmetic applic…

Arithmetic of K3 surfaces defined over finite fields is investigated. In particular, we show that any K3 surface of finite height over a finite field k of characteristic p > 3 has a quasi-canonical lifting to characteristic 0, and that for…

This article classifies Knutsen K3 surfaces all of whose hyperplane sections are irreducible and reduced. As an application, this gives infinite families of K3 surfaces of Picard number two whose general hyperplane sections are…

Following recent work by E. Fuchs et al., we study the Brauer-Manin obstruction for integral points on Wehler K3 surfaces of Markoff type. In particular, we construct some families which fail the integral Hasse principle via the…

Components of the Moduli space of sheaves on a K3 surface are parametrized by a lattice; the (algebraic) Mukai lattice. Isometries of the Mukai lattice often lift to symplectic birational isomorphisms of the collection of components. An…

Based on the result on derived categories on K3 surfaces due to Mukai and Orlov and the result concerning almost-prime numbers due to Iwaniec, we remark the following fact: For any given positive integer N, there are N (mutually…

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…

Let X be a real projective algebraic manifold, s numerates connected components of X(R) and _2Br(X) the subgroup of elements of order 2 of the cohomological Brauer group Br(X). We study the natural homomorphism \xi : _2Br(X) \to (Z/2)^s and…

Let $S$ be a scheme and let $\pi : \mathcal{G} \to S$ 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…

Adapting methods of previous papers by A. Sarti and the author, we construct K3 surfaces from invariants of the Weyl group of type $\Erm_6$. We study in details one of these surfaces, which turns out to have Picard number $20$: for this…

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…

Using equivalences of categories we provide isomorphisms between the Brauer groups of different Hopf algebras. As an example, we show that when k is a field of characteristic different from 2 the Brauer groups BC(k,H_4,r_t) for every dual…

For every supersingular $K3$ surface $X$ in characteristic 2, there exists a homogeneous polynomial $G$ of degree 6 such that $X$ is birational to the purely inseparable double cover of a projective plane defined by $w^2=G$. We present an…

We construct examples of $K3$ surfaces of geometric Picard rank $1$. Our method is a refinement of that of R. van Luijk. It is based on an analysis of the Galois module structure on \'etale cohomology. This allows to abandon the original…

Let $Y$ be a principal homogeneous space of an abelian surface, or a K3 surface, over a finitely generated extension of $\mathbb{Q}$. In 2008, Skorobogatov and Zarhin showed that the Brauer group modulo algebraic classes $\text{Br}\, Y/…

We develop methods of computation of the Brauer-Picard groups of fusion categories and apply them to compute such groups for several classes of fusion categories of prime power dimension: representation categories of elementary abelian…

We study non-isotrivial families of $K3$ surfaces in positive characteristic $p$ whose geometric generic fibers satisfy $\rho\geq21-2h$ and $h\geq3$, where $\rho$ is the Picard number and $h$ is the height of the formal Brauer group. We…

If $K$ is a field of characteristic $p$ then the $p$-torsion of the Brauer group, ${}_p{\rm Br\,}(K)$, is represented by a quotient of the group of $1$-forms, $\Omega^1(K)$. Namely, we have a group isomorphism…

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…

We study the derived categories of twisted supersingular K3 surfaces. We prove a derived crystalline Torelli theorem for twisted supersingular K3 surfaces, characterizing Fourier-Mukai equivalences in terms of isomorphisms between their…

We give a description of the category of ordinary K3 surfaces over a finite field in terms of linear algebra data over Z. This gives an analogue for K3 surfaces of Deligne's description of the category of ordinary abelian varieties over a…