On linear spaces of skew-symmetric matrices of constant rank (Q2487935)

A classical problem in linear algebra is to determine the maximal dimension of a space of matrices (possibly symmetric or skew-symmetric) of constant rank. So far only upper and lower bounds are known [see \textit{B. Ilic} and \textit{J. M. Landsberg}, Math. Ann. 314, 159--174 (1999; Zbl 0949.14028)]. A related question is to characterize the maximal spaces of skew-symmetric matrices of constant rank. The authors' interest in this linear algebra problem is its relation with the study of congruences of lines. In the paper under review linear systems of skew-symmetric matrices of order \(6\) and constant rank \(4\) are considered. In geometrical terms, the authors consider linear subspaces of the dual projective space \(\check {{\mathbb P}}^{14}\) contained in \(\check {{\mathbb G}}(1,5)\), the dual variety of \({\mathbb G}(1,5)\), and not intersecting its singular locus. They classify such linear systems of skew-symmetric matrices of order \(6\) and constant rank \(4\), up to the action of the projective linear group \(\text{PGL}_6\). The main result is that, up to projective automorphisms, there are exactly four types of such linear systems of dimension two, giving four families all of the same dimension.