The geometry of the Coble cubic and orbital degeneracy loci

Vladimiro Benedetti (1) , Laurent Manivel (2) , Fabio Tanturri (3)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 LPP - Laboratoire Paul Painlevé - UMR 8524

The Coble cubics were discovered more than a century ago in connection with genus two Riemann surfaces and theta functions. They have attracted renewed interest ever since. Recently, they were reinterpreted in terms of alternating trivectors in nine variables. Exploring this relation further, we show how the Hilbert scheme of pairs of points on an abelian surface, and also its Kummer fourfold, a very remarkable hyper-Kähler manifold, can very naturally be constructed in this context. Moreover, we explain how this perspective allows us to describe the group law of an abelian surface, in a strikingly similar way to how the group structure of a plane cubic can be defined in terms of its intersection with lines.

Domaines: Mathématiques [math]; Géométrie algébrique [math.AG]

Vladimiro Benedetti, Laurent Manivel, Fabio Tanturri. The geometry of the Coble cubic and orbital degeneracy loci. Mathematische Annalen, 2021, 379 (1-2), pp.415-440. ⟨10.1007/s00208-019-01949-7⟩. ⟨hal-02110007⟩