  Let $K$ be a field and $V$ and $W$ be $K$-vector spaces of dimension $m$ and
$n$. Let $\phi$ be the canonical map from $Hom(V,W)$ to $Hom(\wedge^t
V,\wedge^t W)$. We investigate the Zariski closure $X_t$ of the image $Y_t$ of
$\phi$. In the case $t=\min(m,n)$, $Y_t=X_t$ is the cone over a Grassmannian,
but $X_t$ is larger than $Y_t$ for $1<t<\min(m,n)$. We analyze the
$G=\GL(V)\times\GL(W)$-orbits in $X_t$ via the corresponding $G$-stable prime
ideals. 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 $X_t\setminus Y_t$ 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 $\le 1$ in $Y_t$.
