Alternating trilinear forms on a 9-dimensional space and degenerations of (3,3)-polarized Abelian surfaces

TL;DR

This paper analyzes semisimple elements in the third exterior power of a 9-dimensional space, linking them to Abelian surfaces, their degenerations, and associated genus 2 curves, providing explicit classifications and equations.

Contribution

It classifies degenerations of Abelian surfaces associated with semisimple elements and derives explicit equations for their moduli space.

Findings

Classification of degenerations of Abelian surfaces

Explicit equations for the moduli space

Recovery of genus 2 curves from embeddings

Abstract

We give a detailed analysis of the semisimple elements, in the sense of Vinberg, of the third exterior power of a 9-dimensional vector space over an algebraically closed field of characteristic different from 2 and 3. To a general such element, one can naturally associate an Abelian surface X, which is embedded in 8-dimensional projective space. We study the combinatorial structure of this embedding and explicitly recover the genus 2 curve whose Jacobian variety is X. We also classify the types of degenerations of X that can occur. Taking the union over all Abelian surfaces in Heisenberg normal form, we get a 5-dimensional variety which is a birational model for a genus 2 analogue of Shioda's modular surfaces. We find determinantal set-theoretic equations for this variety and present some additional equations which conjecturally generate the radical ideal.