On the variety of almost commuting nilpotent matrices

Let V be an n-dimensional vector space over an algebraically closed field and n the nilcone of nilpotent endomorphisms of V. We study the variety N = {(X, Y, i, j) ∈ n x n x V x V * | [X, Y] = ij} of n x n of nilpotent matrices with commutator of rank at most one. We describe its irreducible components: two of them correspond to the pairs of commuting matrices, and n - 2 components of smaller dimension corresponding to the pairs of rank one commutator. In our proof we define a map to the zero fiber of the Hilbert scheme of points and study the image and the fibers. We also study the fibers of the map Mnil = {(X, Y, i, j) ∈ n x gl (V) x V x V * | [X,Y] = ij} → n x V, (X, Y, i, j) (X, i) where n x V is the enhanced nilpotent cone.