On the variety of almost commuting nilpotent matrices

Let V be an n-dimensional vector space over an algebraically closed field and $mathcal{N}Let V be an n-dimensional vector space over an algebraically closed field and Nmathcal{N} the nilcone of nilpotent endomorphisms of V. We study the variety A = {(X, Y, i, j) ? N ×N ×V ×V^* |[X, Y] = ij} mathcal{A} = left{{(X, Y, i, j) in mathcal{N} times mathcal{N} times V times V^{ast} vert [X, Y] = ij}right} which is closely related to the variety of pairs of nilpotent n × n matrices with commutator of rank at most 1. 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.