The variety of exterior powers of linear maps

Abstract

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 $Y_t$.