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

TL;DR

This paper classifies the stability of linear systems of skew-symmetric forms on a 6-dimensional space, focusing on those with generic rank up to 4, using geometric invariant theory to analyze group actions.

Contribution

It provides a classification of stable orbits of linear systems of skew-symmetric forms with generic rank 4 or less, advancing understanding of their geometric properties.

Findings

Classification of stable orbits for rank ≤ 4

Analysis of $SL(W)$ action on the projective space

Identification of conditions for GIT stability

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 $\mathbb{P}(\bigwedge^2 W^*)$, are parametrized by the projective space $\mathbb{P}(\mathbb{C}^{n+1}\otimes \bigwedge ^2 W^*)$. 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.