The variety of exterior powers of linear maps

Let V and W be vector spaces of dimension m and n resp. We investigate the Zariski closure Xt of the image Yt of the map HomK(V,W ) → HomK( ∧t V, ∧t W ), φ 7→ ∧t φ . In the case t = min(m,n), Yt = Xt is the cone over a Grassmannian, but for 1 < t < min(m,n) one has Xt 6= Yt . We analyze the G = GL(V )×GL(W )-orbits in Xt via the G-stable prime ideals in O(Xt). 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 Xt \Yt arise from the images Yu for u < t and simple algebraic operations. In the last section we determine the singular locus of Xt . Apart from well-understood exceptional cases, it is formed by the elements of rank ≤ 1 in Yt .