Vector spaces of skew-symmetric matrices of constant rank

TL;DR

This paper classifies the orbits of low-dimensional vector spaces of skew-symmetric matrices with constant rank under group actions and links certain vector bundles on projective planes to matrix spaces.

Contribution

It provides a complete orbit classification for 2-dimensional vector spaces and connects rank two vector bundles on P^2 to skew-symmetric matrices of constant rank.

Findings

Complete description of orbits for 2-dimensional vector spaces.

Existence of matrix spaces corresponding to certain vector bundles.

Relation between vector bundle kernels and matrix ranks.

Abstract

We study the orbits of vector spaces of skew-symmetric matrices of constant rank 2r and type (N+1)x(N+1) under the natural action of SL(N+1), over an algebraically closed field of characteristic zero. We give a complete description of the orbits for vector spaces of dimension 2, relating them to some 1-generic matrices of linear forms. We also show that, for each rank two vector bundle on P^2 defining a triple Veronese embedding of P^2 in G(1,7), there exists a vector space of 8 x 8 skew-symmetric matrices of constant rank 6 whose kernel bundle is the dual of the given rank two vector bundle.