Stable linear systems of skew-symmetric forms of generic rank less than or equal to 4

Abstract

Given a 6-dimensional complex vector space $W$, we consider linear systems of skew-symmetric forms on W. The $n$-dimensional linear systems this kind, that can also be interpreted as $n$-dimensional linear subspaces of $\mathbb{P}({\bigwedge }^2{W}^{\ast })$, are parametrized by the projective space $\mathbb{P}({\mathbb{C}}^{n+1}\otimes {\bigwedge }^2{W}^{\ast })$. We analyze the $SL(W)$ action on this projective space and the GIT stability of linear systems with respect to this action. We present a classification of all stable orbits of linear systems whose generic element is a tensor of rank 4.