On the Brauer Group of an Arithmetic Model of Strictly Complete Intersection in a HyperkÄhler Variety Over a Number Field

2020 ◽  
Vol 250 (1) ◽  
pp. 109-112
Author(s):  
T. V. Prokhorova
Keyword(s):  
Complete Intersection ◽  
Number Field ◽  
Brauer Group

Castelnuovo-Mumford regularity bound for smooth threefolds in ℙ5 and extremal examples

1999 ◽  
Vol 1999 (509) ◽  
pp. 21-34
Author(s):  
Si-Jong Kwak
Keyword(s):  
Complete Intersection ◽  
Number Field ◽  
Complex Number ◽  
Smooth Surfaces ◽  
Veronese Surface ◽  
Optimal Bound ◽  
Integral Curves ◽  
Complex Number Field

Abstract Let X be a nondegenerate integral subscheme of dimension n and degree d in ℙN defined over the complex number field ℂ. X is said to be k-regular if Hi(ℙN, ℐX (k – i)) = 0 for all i ≧ 1, where ℐX is the sheaf of ideals of ℐℙN and Castelnuovo-Mumford regularity reg(X) of X is defined as the least such k. There is a well-known conjecture concerning k-regularity: reg(X) ≦ deg(X) – codim(X) + 1. This regularity conjecture including the classification of borderline examples was verified for integral curves (Castelnuovo, Gruson, Lazarsfeld and Peskine), and an optimal bound was also obtained for smooth surfaces (Pinkham, Lazarsfeld). It will be shown here that reg(X) ≦ deg(X) – 1 for smooth threefolds X in ℙ5 and that the only extremal cases are the rational cubic scroll and the complete intersection of two quadrics. Furthermore, every smooth threefold X in ℙ5 is k-normal for all k ≧ deg(X) – 4, which is the optimal bound as the Palatini 3-fold of degree 7 shows. The same bound also holds for smooth regular surfaces in ℙ4 other than for the Veronese surface.

Descente galoisienne sur le groupe de Brauer

2012 ◽  
Vol 2013 (682) ◽  
pp. 141-165
Author(s):  
Jean-Louis Colliot-Thélène ◽  
Alexei N. Skorobogatov
Keyword(s):  
Number Field ◽  
Characteristic Zero ◽  
Projective Variety ◽  
Algebraic Closure ◽  
Invariant Subgroup ◽  
Brauer Group ◽  
Intersection Pairing ◽  
Nous Montrons ◽  
Numerical Equivalence

Abstract. Soit X une variété projective et lisse sur un corps k de caractéristique zéro. Le groupe de Brauer de X s'envoie dans les invariants, sous le groupe de Galois absolu de k, du groupe de Brauer de la même variété considérée sur une clôture algébrique de k. Nous montrons que le quotient est fini. Sous des hypothèses supplémentaires, par exemple sur un corps de nombres, nous donnons des estimations sur l'ordre de ce quotient. L'accouplement d'intersection entre les groupes de diviseurs et de 1-cycles modulo équivalence numérique joue ici un rôle important. 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 an algebraic closure of k. Under further conditions, e.g., over a number field, we give estimates for the order of this cokernel. We emphasise the rôle played by the exponent of the discriminant groups of the intersection pairing between the groups of divisors and curves modulo numerical equivalence.

THE BRAUER GROUP OF DEL PEZZO SURFACES

2011 ◽  
Vol 07 (02) ◽  
pp. 261-287
Author(s):  
ANDREA C. CARTER
Keyword(s):  
Number Field ◽  
Pezzo Surface ◽  
Del Pezzo Surfaces ◽  
Brauer Group ◽  
Nontrivial Element ◽  
Del Pezzo Surface ◽  
Del Pezzo

Let S1 be a Del Pezzo surface of degree 1 over a number field k. We establish a criterion for the existence of a nontrivial element of order 5 in the Brauer group of S1 in terms of certain Galois-stable configurations of exceptional divisors on this surface.

scholarly journals The Brauer group of cubic surfaces

1993 ◽  
Vol 113 (3) ◽  
pp. 449-460 ◽  
Author(s):  
Sir Peter Swinnerton-Dyer
Keyword(s):  
Finite Group ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Rational Surface ◽  
Hasse Principle ◽  
Weak Approximation ◽  
Brauer Group ◽  
Cubic Surfaces ◽  
Image Position

1. Let V be a non-singular rational surface defined over an algebraic number field k. There is a standard conjecture that the only obstructions to the Hasse principle and to weak approximation on V are the Brauer–Manin obstructions. A prerequisite for calculating these is a knowledge of the Brauer group of V; indeed there is one such obstruction, which may however be trivial, corresponding to each element of Br V/Br k. Because k is an algebraic number field, the natural injectionis an isomorphism; so the first step in calculating the Brauer–Manin obstruction is to calculate the finite group H1 (k), Pic .

HASSE PRINCIPLES FOR ÉTALE MOTIVIC COHOMOLOGY

2018 ◽  
Vol 236 ◽  
pp. 63-83
Author(s):  
THOMAS H. GEISSER
Keyword(s):  
Number Field ◽  
Abelian Varieties ◽  
Base Change ◽  
Hasse Principle ◽  
Brauer Group ◽  
Motivic Cohomology

We discuss the kernel of the localization map from étale motivic cohomology of a variety over a number field to étale motivic cohomology of the base change to its completions. This generalizes the Hasse principle for the Brauer group, and is related to Tate–Shafarevich groups of abelian varieties.

Gyclotomic Division Algebras

1981 ◽  
Vol 33 (5) ◽  
pp. 1074-1084 ◽  
Author(s):  
R. A. Mollin
Keyword(s):  
Finite Group ◽  
Number Field ◽  
Sufficient Conditions ◽  
Algebraic Number Field ◽  
Equivalence Classes ◽  
Necessary And Sufficient Conditions ◽  
Brauer Group ◽  
Division Algebras ◽  
Cyclotomic Algebras ◽  
Necessary And Sufficient

Let K be a field of characteristic zero. The Schur subgroup S(K) of Brauer group B(K) consists of those equivalence classes [A] which contain an algebra which is isomorphic to a simple summand of the group algebra KG for some finite group G. It is well known that the classes in S(K) are represented by cyclotomic algebras, (see [16]). However it is not necessarily the case that the division algebra representatives of these classes are themselves cyclotomic. The main result of this paper is to provide necessary and sufficient conditions for the latter to occur when K is any algebraic number field.Next we provide necessary and sufficient conditions for the Schur group of a local field to be induced from the Schur group of an arbitrary subfield. We obtain a corollary from this result which links it to the main result. Finally we link the concept of the stufe of a number field to the existence of certain quaternion division algebras in S(K).

There are No Transcendental Brauer–Manin Obstructions on Abelian Varieties

2018 ◽  
Vol 2020 (9) ◽  
pp. 2684-2697
Author(s):  
Brendan Creutz
Keyword(s):  
Abelian Variety ◽  
Number Field ◽  
Abelian Varieties ◽  
Rational Points ◽  
Brauer Group

Abstract Suppose X is a torsor under an abelian variety A over a number field. We show that any adelic point of X that is orthogonal to the algebraic Brauer group of X is orthogonal to the whole Brauer group of X. We also show that if there is a Brauer–Manin obstruction to the existence of rational points on X, then there is already an obstruction coming from the locally constant Brauer classes. These results had previously been established under the assumption that A has finite Tate–Shafarevich group. Our results are unconditional.

Cyclotomic Splitting Fields

1982 ◽  
Vol 25 (2) ◽  
pp. 222-229 ◽  
Author(s):  
R. A. Mollin
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Sufficient Conditions ◽  
Algebraic Number Field ◽  
Abelian Extension ◽  
Brauer Group ◽  
Division Algebras ◽  
Necessary And Sufficient ◽  
Cyclotomic Extension ◽  
Maximal Subfield

AbstractLet D be a division algebra whose class [D] is in B(K), the Brauer group of an algebraic number field K. If [D⊗KL] is the trivial class in B(L), then we say that L is a splitting field for D or L splits D. The splitting fields in D of smallest dimension are the maximal subfields of D. Although there are infinitely many maximal subfields of D which are cyclic extensions of K; from the perspective of the Schur Subgroup S(K) of B(K) the natural splitting fields are the cyclotomic ones. In (Cyclotomic Splitting Fields, Proc. Amer. Math. Soc. 25 (1970), 630-633) there are errors which have led to the main result of this paper, namely to provide necessary and sufficient conditions for (D) in S(K) to have a maximal subfield which is a cyclic cyclotomic extension of K, a finite abelian extension of Q. A similar result is provided for quaternion division algebras in B(K).

Sign in / Sign up

Export Citation Format

Share Document