Algebra & Number Theory
- Algebra Number Theory
- Volume 9, Number 3 (2015), 629-679.
Fano schemes of determinants and permanents
Let and denote the subschemes of given by the determinants (respectively the permanents) of an matrix of indeterminates. In this paper, we study the geometry of the Fano schemes and parametrizing the -dimensional planes in lying on and , 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 always has the expected dimension, and we describe its components exactly. Finally, we give a detailed study of the Fano schemes of -planes on the determinantal and permanental hypersurfaces.
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
Permanent link to this document
Digital Object Identifier
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]
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. https://projecteuclid.org/euclid.ant/1510842314