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

Abstract

We give a detailed analysis of the semisimple elements, in the sense of Vinberg, of the third exterior power of a nine-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 eight-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 five-dimensional variety which is a birational model for a genus 2 analog 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.