Algebra & Number Theory

Fano schemes of determinants and permanents

Melody Chan and Nathan Ilten

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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.

Article information

Algebra Number Theory, Volume 9, Number 3 (2015), 629-679.

Received: 10 June 2014
Revised: 15 January 2015
Accepted: 23 February 2015
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14M12: Determinantal varieties [See also 13C40]
Secondary: 14N20: Configurations and arrangements of linear subspaces 14C05: Parametrization (Chow and Hilbert schemes) 15A15: Determinants, permanents, other special matrix functions [See also 19B10, 19B14] 14B10: Infinitesimal methods [See also 13D10]

Fano schemes determinantal varieties permanent


Chan, Melody; Ilten, Nathan. Fano schemes of determinants and permanents. Algebra Number Theory 9 (2015), no. 3, 629--679. doi:10.2140/ant.2015.9.629.

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