The Enriques Conjectures

Two fundamental facts about the moduli space M_g of smooth curves of genus g are what are called Harer’s theorems: that the Picard group of M_g is of rank one, generated (over the rational numbers) by the Hodge class; and that the relative Picard group of the universal curve over M_g is also of rank one, generated by the relative dualizing sheaf. We can make analogous statements about the Severi variety of plane curves and the Hurwitz space parametrizing branched covers, which are still open; in fact, the former was conjectured by Enriques more than a century ago and remains open.

In this talk I’d like to describe all of these theorems/conjectures, and the implications among them, including Isabel Vogt’s recent work on Severi varieties. I’ll be working entirely with rational coefficients, so torsion classes, which are far more mysterious, will not enter into it.

For more information, please see https://researchseminars.org/seminar/harvard-mit-ag-seminar