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