Related papers: Sur le groupe de Brauer transcendant

Let $K/\mathbb{Q}$ be a finitely generated field of characteristic zero and $X/K$ a smooth projective variety. Fix $q\in\mathbb{N}$. For every prime number $\ell$ let $\rho_\ell$ be the representation of $\mathrm{Gal}(K)$ on the \'etale…

Let $k$ be a field of characteristic zero and ${\bar k}$ an algebraic closure of $k$. For a geometrically integral variety $X$ over $k$, we write ${\bar k}(X)$ for the function field of ${\bar X}=X\times_k{\bar k}$. If $X$ has a smooth…

For varieties given by an equation N_{K/k}(\Xi)=P(t), where N_{K/k} is the norm form attached to a field extension K/k and P(t) in k[t] is a polynomial, three topics have been investigated: (1) computation of the unramified Brauer group of…

We prove that the $p^\infty$-torsion of the transcendental Brauer group of an abelian variety over a finitely generated field of characteristic $p>0$ is bounded. This answers a (variant of a) question asked by Skorobogatov and Zarhin for…

Let $A$ be an abelian variety over a field finitely generated over $\mathbb{Q}$. We show that the finiteness of the $\ell$-primary torsion subgroup of the higher Brauer group is a sufficient criterion for the Tate conjecture to hold.…

Let $K$ be the function field of a smooth curve over an algebraically closed field $k$. Let $X$ be a scheme, which is smooth and projective over $K$. Suppose that the cotangent bundle $\Omega_{X/K}$ is ample. Let $R:={\rm Zar}(X)(K)\cap X)$…

We prove that a smooth proper universally CH_0-trivial variety X over a field k has universally trivial Brauer group. This fills a gap in the literature concerning the p-torsion of the Brauer group when k has characteristic p.

We 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 Enr(X) of Enriques quotients of X up to isomorphism…

We present a conjectural formula describing the cokernel of the Albanese map of zero-cycles of smooth projective varieties $X$ over $p$-adic fields in terms of the N\'eron-Severi group and provide a proof under additional assumptions on an…

Let G be a connected algebraic group over an algebraically closed field of characteristic p (possibly 0), and X a variety on which G acts transitively with connected stabilizers. We show that any \'etale Galois cover of X of degree prime to…

In this paper we show that the l^n-torsion part of the cohomological Brauer groups of certain schemes associated to symmetric powers of a projective smooth curve over a separably closed field k are isomorphic, when `l is invertible in k.…

We show that Colliot-Th\'el\`ene's conjecture on 0-cycles of degree 1 implies finiteness for the u-invariant of the function field of a curve over a totally imaginary number field and period-index bounds for the Brauer groups of arbitrary…

We study the Brauer groups of affine surfaces that are complements of singular hyperplane sections of smooth cubic surfaces over a field $k$ of characteristic $0$. We determine the Brauer group over the algebraic closure as a Galois module…

Let H be a homology theory for algebraic varieties over a field k. To a complete k-variety X, one naturally attaches an ideal of the coefficient ring H(k). We show that, when X is regular, this ideal depends only on the upper Chow motive of…

Let $X$ be an irreducible smooth projective curve, defined over an algebraically closed field $k$, of genus at least three and $L$ a line bundle on $X$. Let ${\mathcal M}_X(r,L)$ be the moduli space of stable vector bundles on $X$ of rank…

We obtain an easy sufficient condition for the Brauer group of a diagonal quartic surface D over Q to be algebraic. We also give an upper bound for the order of the quotient of the Brauer group of D by the image of the Brauer group of Q.…

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…

Let $X$ be the surface $\T\times\T$ where $\T$ is the complex torus. This paper is the third in a series, studying the fundamental group of the Galois cover of $X$ \wrt a generic projection onto $\C\P^2$. Van Kampen Theorem gives a…

Unlike the classical Brauer group of a field, the Brauer-Grothendieck group of a singular scheme need not be torsion. We show that there exist integral normal projective surfaces over a large field of positive characteristic with…

Let $L$ be a number field and let $E/L$ be an elliptic curve with complex multiplication by the ring of integers $\mathcal{O}_K$ of an imaginary quadratic field $K$. We use class field theory and results of Skorobogatov and Zarhin to…