#### Vol. 9, No. 3, 2015

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editors' Interests Submission Guidelines Submission Form Editorial Login Ethics Statement Author Index To Appear ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Other MSP Journals
Fano schemes of determinants and permanents

### Melody Chan and Nathan Ilten

Vol. 9 (2015), No. 3, 629–679
##### Abstract

Let ${D}_{m,n}^{r}$ and ${P}_{m,n}^{r}$ denote the subschemes of ${ℙ}^{mn-1}$ given by the $r×r$ determinants (respectively the $r×r$ permanents) of an $m×n$ 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 ${ℙ}^{mn-1}$ 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 $3×3$ determinantal and permanental hypersurfaces.

##### Keywords
Fano schemes, determinantal varieties, permanent
##### Mathematical Subject Classification 2010
Primary: 14M12
Secondary: 14N20, 14C05, 15A15, 14B10
##### Milestones
Received: 10 June 2014
Revised: 15 January 2015
Accepted: 23 February 2015
Published: 17 April 2015
##### Authors
 Melody Chan Department of Mathematics Harvard University Cambridge, MA 02138 United States Nathan Ilten Department of Mathematics Simon Fraser University Burnaby BC V5A1S6 Canada