The Brauer group, classifying Azumaya algebras up to Morita equivalence, is a fundamental invariant in number theory and algebraic geometry. Given a moduli problem M (e.g. smooth curves of a given genus, K3 surfaces, abelian varieties of a given dimension…) one can consider an element of the Brauer group of M as a way to functorially assign to any family X -> S in M(S) an element in the Brauer group of S.

If we consider the moduli problem M_g of smooth curves of a given genus, the Brauer groups of M_{1,1} (the moduli problem of elliptic curves) and M_2 are known over a vast generality of bases, for example Br(M_{1,1}) is known when the base is any field or the integers; the Brauer group of M_g for g at least 4 is known to be trivial over the complex numbers through topological methods. The case g=3 was until recently open over any base.

In a recent paper with Andrea di Lorenzo (Università di Pisa) we show that over any field k of characteristic zero the Brauer group of M_3 is equal to a direct sum of Br(k) and a copy of Z/2Z. To our surprise, the proof of this result goes through one of the most well-known theorems in classical enumerative geometry: there are exactly 27 lines lying on a cubic surface in P^3.