Stable linear systems of skew-symmetric forms of generic rank \(\leq 4\) (Q2029826)

Consider a complex vector space $W$ of dimension $6$, an $(n+1)$-dimension complex vector space $V$ and the $n$-dimension linear system of skew symmetric forms on $W$ that define a linear embedding $\mathbb{P}(V)\rightarrow\mathbb{P}(\bigwedge^2W^*)$. In the present paper the author discusses the orbits of the $\mathrm{SL}(W)$ group action on $\mathbb{P}(V^*\otimes \bigwedge^2W^*)$. For background see [\textit{E. Ferapontov} and \textit{L. Manivel}, ``On a class of spaces of skew-symmetric forms related to Hamiltonian systems of conservation laws'', Preprint, \url{arXiv:1810.12216}] and [\textit{L. Manivel} and \textit{E. Mezzetti}, Manuscr. Math. 117, No. 3, 319--331 (2005; Zbl 1084.14050)]. The author, being interested in the construction of moduli spaces , concentrates on stable orbits in the sense of Mumford's Geometric Invariant Theory. The author gives a complete classification of stable orbits of linear systems of rank $r$ $\leq4$. Furthermore, the author ends the paper with an explanation of the role that 4-dimensional linear systems of skew-symmetric forms play in the study of moduli spaces of sheaves on cubic threefolds. See also [\textit{M. D. Atkinson}, J. Aust. Math. Soc., Ser. A 34, 306--315 (1983; Zbl 0521.15009)]