ON LINEAR SUBSPACES OF Mn AND THEIR SINGULAR SETS RELATED TO THE CHARACTERISTIC MAP

We study linear subspaces L ⊆ Mn (over an algebraically closed field F of characteristic zero) and their singular sets S(L) defined by S(L) = {A ∈ Mn : χ(A+ L) is not dense in Fn}, where χ : Mn −→ F n is the characteristic map. We give a complete characterization of the subspaces L ⊂ M2 such that ∅ 6= S(L) 6= M2. We also provide a complete characterization of the singular sets S(L) in the case of n = 2. Finally, we give a characterization of the n-dimensional subspaces L ⊂ Mn such that S(L) = ∅ by means of their intersections with conjugacy classes. 1. Preliminaries and introduction We work throughout over an algebraically closed field F of characteristic zero. We define F = F \ {0}. We denote by #E the cardinality of a finite set E. The set of all (n × n)-matrices whose entries are elements of F is denoted by Mn. (We assume throughout that n ≥ 2.) The zero matrix and the unit matrix belonging to Mn are denoted by O and I, respectively. We define GLn to be the full linear group of size n over the field F, i. e. GLn = {U ∈ Mn : det(U) 6= 0}. The conjugacy class of a matrix A ∈ Mn is denoted by O(A). (In other words, O(A) = {UAU : U ∈ GLn}.) A subset E ⊆ Mn is said to be triangularizable if there is a U ∈ GLn such that UEU := {UAU : A ∈ E} consists of upper triangular matrices. The subset E is said to be GLn-invariant if UEU ⊆ E for all U ∈ GLn. We consider F, Mn ∼= F n , and their subsets as topological spaces endowed with the Zariski topology. We say that a property holds for a generic 2000 Mathematics Subject Classification. 15A18, 14A10, 14L35.