Fano schemes of determinants and permanents

Abstract

Let $D_{m,n}^r$ and $P_{m,n}^r$ denote the subschemes of ${\mathbb{P}}^{mn-1}$ given by the $r\times r$ determinants (respectively the $r\times r$ permanents) of an $m\times n$ matrix of indeterminates. In this paper, we study the geometry of the Fano schemes $F_k\left(D_{m,n}^r\right)$ and $F_k\left(P_{m,n}^r\right)$ parametrizing the $k$-dimensional planes in ${\mathbb{P}}^{mn-1}$ lying on $D_{m,n}^r$ and $P_{m,n}^r$, respectively. We prove results characterizing which of these Fano schemes are smooth, irreducible, and connected; and we give examples showing that they need not be reduced. We show that $F_1\left(D_{n,n}^n\right)$ always has the expected dimension, and we describe its components exactly. Finally, we give a detailed study of the Fano schemes of $k$-planes on the $3\times 3$ determinantal and permanental hypersurfaces.