The variety of exterior powers of linear maps

Abstract

Let V and W be vector spaces of dimension m and n respectively. We investigate the Zariski closure X(t) of the image Y(t) of the map Hom(K) (V, W) -> Hom(K) (boolean AND(t) V, boolean AND(t) W), phi -> boolean AND(t) phi. In the case t = min(m, n), Y(t) = X(t) is the cone over a Grassmannian, but for 1 < t < min(m, n) one has X(t) not equal Y(t). We analyze the G = GL(V) x GL(W)-orbits in X(t) via the G-stable prime ideals in O(X(t)). It turns out that they are classified by two numerical invariants, one of which is the rank and the other a related invariant that we call small rank. Surprisingly, the orbits in X(t)Y(t) arise from the images Y(u) for u < t and simple algebraic operations. In the last section we determine the singular locus of X(t). Apart from well-understood exceptional cases, it is formed by the elements of rank <= 1 in Y(t). (C) 2008 Elsevier Inc. All rights reserved.