Brauer groups of moduli of hyperelliptic curves via cohomological invariants

Using the theory of cohomological invariants for algebraic stacks, we compute the Brauer group of the moduli stack of hyperelliptic curves ${\mathcal {H}}_g$ over any field of characteristic $0$ . In positive characteristic, we compute the part of the Brauer group whose order is prime to the characteristic of the base field.