Brauer groups and obstruction problems: moduli spaces and arithmetic

February 25 to March 1, 2013

at the

American Institute of Mathematics, San Jose, California

organized by

Asher Auel, Brendan Hassett, Anthony V'arilly-Alvarado, and Bianca Viray

Original Announcement

This workshop will be devoted to studying elements of the Brauer group from both an arithmetic perspective and a Hodge-theoretic and derived categorical perspective.

Elements of the Brauer group arise in the theory of obstructions to the existence of rational points on varieties and to the existence of universal objects of moduli spaces of stable sheaves. The goal of this workshop is to bring together researchers in number theory, complex algebraic geometry, derived categories, and Hodge theory to study connections between these perspectives on Brauer group elements.

The main topics for the workshop are

  1. Brauer classes arising in twisted derived equivalence as obstructions to rational points.
  2. Transcendental Brauer classes arising in Brauer--Manin obstructions as providing twisted derived equivalences.
  3. Explicitly representing Brauer classes as symbols and via non-fine moduli spaces of stable objects.
There has been a flurry of activity in recent years on semi-orthogonal decompositions of the derived category, twisted derived equivalences, the theory of stability conditions, and the birational geometry of cubic hypersurfaces. Questions central to the workshop concern the investigation of well-known cases of twisted derived equivalences and semiorthogonal decomposition (Kummer surfaces, K3 surfaces, elliptic threefolds, cubic threefolds, conic bundles over rational surfaces, abelian varieties, quadric fibrations, and quadric intersection fibrations) for instances providing Brauer--Manin obstructions. Do Brauer--Manin obstructions exist when purely Hodge-theoretic criteria provide nontrivial transcendental Brauer classes? Can specific arithmetic techniques be brought to bear in constructing examples of new twisted derived equivalences?

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

Abelian n-division fields of elliptic curves and Brauer groups of product Kummer & abelian surfaces
by  Anthony Várilly-Alvarado and Bianca Viray,  Forum Math. Sigma 5 (2017), e26, 42 pp.  MR3731278
On a local-global principle for H3 of function fields of surfaces over a finite field
by  Alena Pirutka
ACM bundles on cubic fourfolds containing a plane
by  Marti Lahoz, Emanuele Macri and Paolo Stellari
Universal spaces for unramified Galois cohomology
by  Fedor Bogomolov and Yuri Tschinkel
Derived equivalences and rational points of twisted K3 surfaces
by  Kenneth Ascher, Krishna Dasaratha, Alexander Perry, and Rong Zhou,  Brauer groups and obstruction problems, 13–28, Progr. Math., 320, Birkhäuser/Springer, Cham, 2017  MR3616005
Twisted derived equivalences for affine schemes
by  Benjamin Antieau,  Brauer groups and obstruction problems, 7–12, Progr. Math., 320, Birkhäuser/Springer, Cham, 2017  MR3616004
The Brauer group is not a derived invariant
by  Nicolas Addington
Computing Néron-Severi groups and cycle class groups
by  Bjorn Poonen, Damiano Testa and Ronald van Luijk,  Compos. Math. 151 (2015), no. 4, 713-734  MR3334893
Brauer groups on K3 surfaces and arithmetic applications
by  by Kelly McKinnie, Justin Sawon, Sho Tanimoto, and Anthony Várilly-Alvarado,  Brauer groups and obstruction problems, 177–218,Progr. Math., 320, Birkhäuser/Springer, Cham, 2017  MR3616011
Moriwaki divisors and the augmented base loci of divisors on the moduli space of curves
by  Salvatore Cacciola, Angelo Felice Lopez and Filippo Viviani,  Michigan Math. J. 65 (2016), no. 3, 533-546  MR3542764
Unramified Brauer classes on cyclic covers of the projective plane
by  Colin Ingalls, Andrew Obus, Ekin Ozman, and Bianca Viray
Universal unramified cohomology of cubic fourfolds containing a plane
by  Asher Auel, Jean-Louis Colliot-Th�l�ne and R. Parimala,  Brauer groups and obstruction problems, 29–55, Progr. Math., 320, Birkhäuser/Springer, Cham, 2017  MR3616006
Descente galoisienne sur le second groupe de Chow : mise au point
by  Jean-Louis Colliot-Th�l�ne,  Doc. Math. 2015, Extra vol.: Alexander S. Merkurjev's sixtieth birthday, 195-220  MR3404380
Hypersurfaces quartiques de dimension 3 : non rationalit� stable
by  Jean-Louis Colliot-Th�l�ne and Alena Pirutka,  Ann. Sci. Éc. Norm. Supér. (4) 49 (2016), no. 2, 371-397  MR3481353
Unirational threefolds with no universal codimension 2 cycle
by  Claire Voisin,  Invent. Math. 201 (2015), no. 1, 207-237  MR3359052