Algebraic and Arithmetic Geometry Seminar

# Brauer groups of moduli of hyperelliptic curves, via cohomological invariants

## by Roberto Pirisi (KTH Stockholm)

Europe/Rome
Aula Magna (Dipartimento di Matematica)

### Aula Magna

#### Dipartimento di Matematica

Description

We use the theory of cohomological invariants for algebraic stacks to completely describe the Brauer group of the moduli stacks $H_g$ of genus $g$ hyperellitic curves over fields of characteristic zero, and the prime-to-$\mathsf{char}(k)$ part in positive characteristic. It turns out that the (non-trivial part of the) group is generated by cyclic algebras, by 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. This is joint work with Andrea di Lorenzo.