Vol. 9, No. 3, 2015

Download this article
Download this article For screen
For printing
Recent Issues

Volume 11
Issue 5, 1009–1241
Issue 4, 767–1007
Issue 3, 505–765
Issue 2, 253–503
Issue 1, 1–252

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
Editorial Board
Editors' Addresses
Editors' Interests
About the Journal
Scientific Advantages
Submission Guidelines
Submission Form
Editorial Login
Author Index
To Appear
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Fano schemes of determinants and permanents

Melody Chan and Nathan Ilten

Vol. 9 (2015), No. 3, 629–679

Let Dm,nr and Pm,nr denote the subschemes of mn1 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 Fk(Dm,nr) and Fk(Pm,nr) parametrizing the k-dimensional planes in mn1 lying on Dm,nr and Pm,nr, 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 F1(Dn,nn) 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.

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