# The geometry of the Coble cubic and orbital degeneracy loci

## TL;DR

This paper explores the geometric structure of Coble cubics, connecting them to abelian surfaces, Kummer fourfolds, and their group laws, providing new insights into classical algebraic geometry through modern reinterpretations.

## Contribution

It introduces a novel perspective linking Coble cubics with abelian surfaces and hyper-Kähler manifolds, elucidating their geometric and group-theoretic properties.

## Key findings

- Constructs the Hilbert scheme of pairs of points on an abelian surface from Coble cubics
- Describes the Kummer fourfold as a hyper-Kähler manifold within this framework
- Provides a new geometric interpretation of the group law on abelian surfaces

## Abstract

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-K\"ahler 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.