  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.
