Related papers: On the Brauer groups of fibrations II

Let $\mathcal{X}\rightarrow C$ be a dominant morphism between smooth irreducible varieties over a finitely generated field $k$ such that the generic fiber $X$ is smooth, projective and geometrically connected. Assuming that $C$ is a curve…

Let $\mathcal X$ be a regular variety, flat and proper over a complete regular curve over a finite field, such that the generic fiber $X$ is smooth and geometrically connected. We prove that the Brauer group of $\mathcal X$ is finite if and…

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…

This is an English translation of the author's 1989 note in Russian, published in a collection "Arithmetic and Geometry of Varieties" (V.E. Voskresenski, ed.), Kuibyshev State University, Kuibyshev, 1989, pp. 57--67. Let $X$ be be an…

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…

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…

We give a formula relating the order of the Brauer group of a surface fibered over a curve over a finite field to the order of the Tate-Shafarevich group of the Jacobian of the generic fiber. The formula implies that the Brauer group of a…

Let $X$ be a rationally connected algebraic variety, defined over a number field $k$. We find a relation between the arithmetic of rational points on $X$ and the arithmetic of zero-cycles. More precisely, we consider the following…

Let X be a proper smooth and connected surface over a finite field. We proved in [LLR2] that the order of the Brauer group Br(X) of X is a perfect square if it is finite. Our proof is based in part on a result of Gordon [Gor], which we used…

Let $X$ be a smooth projective surface over a number field $K$. Assume that $X$ has an elliptic fibration over $\mathbb{P}^1_K$ with at least one singular fibre and a section. Let $\mathcal{X}/U$ be a smooth projective model of $X$ over…

Let $X$ be a smooth projective integral variety over a finitely generated field $k$ of characteristic $p>0$. We show that the finiteness of the exponent of the $p$-primary part of $\mathrm{Br}(X_{k^s})^{G_k}$ is equivalent to the Tate…

Fix an $I$-adically complete Noetherian ring $A$ and suppose $X$ is a proper $A$-scheme. This article concerns the relationship between the Brauer group of $X$ and that of the various $X_n$ where $X_n$ is the fiber over $A/I^{n+1}$. In…

In this paper we describe the fundamental group-scheme of a proper variety fibered over an abelian variety with rationally connected fibers over an algebraically closed field. We use old and recent results for the Nori fundamental…

Let $k$ be a non-archimedean complete field. We prove a substitute for the reduced fiber theorem (of Bosch, L\"utkebohmert and Raynaud) that holds for every morphism $Y\to X$ flat and with geometrically reduced fibers between $k$-affinoid…

Let $k$ be a field finitely generated over the finite field $\mathbb F_p$ of odd characteristic $p$. For any K3 surface $X$ over $k$ we prove that the prime to $p$ component of the cokernel of the natural map $Br(k)\to Br(X)$ is finite.

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…

Extending a result of Schr\"oer on a Grothendieck question in the context of complex analytic spaces, we prove that the surjectivity of the Brauer map $\delta: Br(X) \rightarrow H_{\rm \'et}^2(X,\mathbb{G}_{m, X})_{\rm tor}$ for algebraic…

Let $G$ be one of the finite general linear, unitary, symplectic or orthogonal groups over finite fields of odd order. We find the cardinality of the fibers of the square map at a given generic element. Using this we find the number of real…

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…

Let X be a smooth and proper variety over a number field k. Conjectures on the image of the Chow group of zero-cycles of X in the product of the corresponding groups over all completions of k were put forward by Colliot-Th\'el\`ene, Kato…