Abstract
Given an algebraic variety X, the Brauer group
of X is the group of Azumaya algebras over X, or equivalently
the group of Severi-Brauer varieties over X, i.e. fibrations
over X which are étale locally isomorphic to a projective space.
It was first studied in the case where X is the spectrum of a
field by Noether and Brauer, and has since became a central
object in algebraic and arithmetic geometry, being for example
one of the first obstructions to rationality used to produce
counterexamples to Noether's problem of whether given a
representation V of a finite group G the quotient V/G is
rational. While the Brauer group has been widely studied for
schemes, computations at the level of moduli stacks are
relatively recent, the most prominent of them being the
computations by Antieau and Meier of the Brauer group of the
moduli stack of elliptic curves over a variety of bases,
including Z, Q, and finite fields.
In a recent series of joint works with A. Di Lorenzo, we use the
theory of cohomological invariants, and its extension to
algebraic stacks, to completely describe the Brauer group of the
moduli stacks of hyperelliptic curves, and their
compactifications, over fields of characteristic zero, and the
prime-to-char(k) part in positive characteristic. It turns out
that the Brauer group of the non-compact stack is generated by
elements coming from the base field, cyclic algebras, an element
coming from a map to the classifying stack of étale algebras of
degree 2g+2, and when g is odd by the Brauer-Severi fibration
induced by taking the quotient of the universal curve by the
hyperelliptic involution. This paints a richer picture than in
the case of elliptic curves, where all non-trivial elements come
from cyclic algebras. Regarding the compactifications, there are
two natural ones, the first obtained by taking stable
hyperelliptic curves and the second by taking admissible covers.
It turns out that the Brauer group of the former is trivial,
while for the latter it is almost as large as in the non-compact
case, a somewhat surprising difference as the two stacks are
projective, smooth and birational, which would force their
Brauer groups to be equal if they were schemes.