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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.

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