Vol. 9, No. 3, 2015

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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