Vector spaces of skew-symmetric matrices of constant rank

Abstract

We study the orbits of vector spaces of skew-symmetric matrices of constant rank 2 r and type ( N + 1 ) × ( 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 × 8 skew-symmetric matrices of constant rank 6 whose kernel bundle is the dual of the given rank two vector bundle.