Brauer groups and obstruction problems: moduli spaces and arithmetic
February 25 to March 1, 2013
American Institute of Mathematics,
San Jose, California
and Bianca Viray
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
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?
- Brauer classes arising in twisted derived equivalence as
obstructions to rational points.
- Transcendental Brauer classes arising in Brauer--Manin obstructions
as providing twisted derived equivalences.
- Explicitly representing Brauer classes as symbols and via non-fine
moduli spaces of stable objects.
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