A note on linear subspaces of determinantal varieties. (English) Zbl 0861.14045

\(V\) and \(W\) denote finite dimensional vector spaces over an algebraically closed field of characteristic zero. Within the scope of a general classification of subspaces \(M \subseteq \operatorname{Hom} (V,W)\) the author studies particularly vector spaces \(M\) in a determinantal variety in \(\operatorname{Hom} (V,W)\), i.e. those \(M\) satisfying the condition \(\text{rank} \varphi\leq r\) for every \(\varphi\in M\) with \(0<r< \max (\dim V, \dim W)\), \(r\) fixed. In this context he presents a condition on the dimension of \(M\) which implies that \(M\) belongs to a distinguished family of low rank subspaces (so-called compression spaces, see corollary 2). Moreover he gives a necessary condition on \(M\) \((\dim M \leq r^2)\) to be a so-called primitive space (see corollary 3).
Reviewer: M.Herrmann (Köln)


14M12 Determinantal varieties
13C40 Linkage, complete intersections and determinantal ideals