Fano schemes of determinants and permanents

Chan, Melody; Ilten, Nathan

doi:10.2140/ant.2015.9.629

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Fano schemes of determinants and permanents

Melody Chan and Nathan Ilten

Vol. 9 (2015), No. 3, 629–679

DOI: 10.2140/ant.2015.9.629

Abstract

Let $D_{m, n}^{r}$ and $P_{m, n}^{r}$ denote the subschemes of $ℙ^{m n - 1}$ given by the $r \times r$ determinants (respectively the $r \times r$ permanents) of an $m \times n$ matrix of indeterminates. In this paper, we study the geometry of the Fano schemes $F_{k} (D_{m, n}^{r})$ and $F_{k} (P_{m, n}^{r})$ parametrizing the $k$ -dimensional planes in $ℙ^{m n - 1}$ lying on $D_{m, n}^{r}$ and $P_{m, n}^{r}$ , 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 $F_{1} (D_{n, n}^{n})$ 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 $3 \times 3$ determinantal and permanental hypersurfaces.

Keywords

Fano schemes, determinantal varieties, permanent

Mathematical Subject Classification 2010

Primary: 14M12

Secondary: 14N20, 14C05, 15A15, 14B10

Milestones

Received: 10 June 2014

Revised: 15 January 2015

Accepted: 23 February 2015

Published: 17 April 2015

Authors

Melody Chan
Department of Mathematics Harvard University Cambridge, MA 02138 United States

Nathan Ilten
Department of Mathematics Simon Fraser University Burnaby BC V5A1S6 Canada

Vol. 9, No. 3, 2015