Conjugacy classes of commuting nilpotents

We consider the space M_q,n of regular q-tuples of commuting nilpotent endomorphisms of kⁿ modulo simultaneous conjugation. We show that M_q,n admits a natural homogeneous space structure, and that it is an affine space bundle over Pq−¹. A closer look at the homogeneous structure reveals that, over C and with respect to the complex topology, M_q,n is a smooth vector bundle over Pq−¹. We prove that, in this case, M_q,n is diffeomorphic to a direct sum of twisted tangent bundles. We also prove that M_q,n possesses a universal property and represents a functor of ideals, and we use it to identify M_q,n with an open subscheme of a punctual Hilbert scheme. Using a result of A. Iarrobino’s, we show that M_q,n → Pq−¹ is not a vector bundle, hence giving a family of affine space bundles that are not vector bundles.