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2015 Fano schemes of determinants and permanents
Melody Chan, Nathan Ilten
Algebra Number Theory 9(3): 629-679 (2015). DOI: 10.2140/ant.2015.9.629

Abstract

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.

Citation

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Melody Chan. Nathan Ilten. "Fano schemes of determinants and permanents." Algebra Number Theory 9 (3) 629 - 679, 2015. https://doi.org/10.2140/ant.2015.9.629

Information

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

zbMATH: 1329.14101
MathSciNet: MR3340547
Digital Object Identifier: 10.2140/ant.2015.9.629

Subjects:
Primary: 14M12
Secondary: 14B10, 14C05, 14N20, 15A15

Rights: Copyright © 2015 Mathematical Sciences Publishers

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