Roberto Pirisi: Brauer groups of moduli problems via enumerative geometry

Abstract:

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, quasi polarized K3 surfaces of a given degree, principally polarized 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 \(X\to S\) 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.

I will discuss my work with Andrea di Lorenzo (Università di Pisa) where 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 \(\mathbb{Z}/2\mathbb{Z}\). 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 \(\mathbb{P}^3\). I will also talk of recent work with Andrés Jaramillo Puentes (Universität Tübingen) where we expanded these techniques to compute the pullback of the Brauer group of \(M_3\) to the moduli stack of genus three hyperelliptic curves \(\mathcal{H}_3\) and produce for all \(g\) new mod-2 étale cohomology classes on \(M_g\) in degree \(2^{g-2}\) and \(2^{g-1}\).