The variety of exterior powers of linear maps
| dc.creator | Bruns, Winfried | |
| dc.creator | Conca, Aldo | |
| dc.date | 2007-05-23 | |
| dc.date | 2008-04-02 | |
| dc.date.accessioned | 2026-07-07T09:29:31Z | |
| dc.date.available | 2026-07-07T09:29:31Z | |
| dc.description | Let $K$ be a field and $V$ and $W$ be $K$-vector spaces of dimension $m$ and $n$. Let $ϕ$ 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 $ϕ$. 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 $Y_t$. | |
| dc.description | Few minor changes. Final version to appear in J. of Algebra | |
| dc.identifier | https://arxiv.org/abs/0705.3399 | |
| dc.identifier | http://arxiv.org/abs/0705.3399 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/157810 | |
| dc.subject | Commutative Algebra | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 13A50, 14L30 | |
| dc.title | The variety of exterior powers of linear maps | |
| dc.type | text |