Algebraic and Arithmetic Geometry Seminar

Brauer groups of moduli of hyperelliptic curves, via cohomological invariants

by Roberto Pirisi (KTH Stockholm)

Aula Magna (Dipartimento di Matematica)

Aula Magna

Dipartimento di Matematica


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.