Nets of alternating matrices and the linear syzygy conjectures (Q1333231)

For a connected linear group acting on a linear space of matrices in such a way that every level set for the invariants contains finitely many orbits, there are only a few possibilities. They have been determined by \textit{V. G. Kac} [J. Algebra 64, 190-213 (1980; Zbl 0431.17007)]. The authors carry out a complete classification of two of those cases: 3- dimensional spaces (nets) of alternating \(4 \times 4\) matrices and 3- dimensional spaces of alternating \(5 \times 5\) matrices. They apply the second classification to prove a special case of their linear syzygy conjecture. Meanwhile, a proof of the general case of this conjecture has been announced by \textit{Mark Green} [The Eisenbud-Koh-Stillman conjecture, preprint (1993)].