The variety of exterior powers of linear maps

TL;DR

This paper studies the geometric structure of the Zariski closure of the images of exterior power maps of linear transformations, revealing orbit classifications and singular locus characteristics.

Contribution

It characterizes the orbit structure of the Zariski closure of exterior power images and introduces the concept of small rank as a key invariant.

Findings

Orbits are classified by rank and small rank invariants.

The Zariski closure $X_t$ contains images from lower-dimensional cases.

The singular locus is mainly elements of rank ≤ 1 in $Y_t$, except for special cases.

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 $Y_t$.