Conjugacy classes of commuting nilpotents

Abstract

We consider the space ${\mathcal{M}}_{q,n}$ of regular $q$-tuples of commuting nilpotent endomorphisms of $k^n$ modulo simultaneous conjugation. We show that ${\mathcal{M}}_{q,n}$ admits a natural homogeneous space structure, and that it is an affine space bundle over ${\mathbb{P}}^{q-1}$. A closer look at the homogeneous structure reveals that, over $\mathbb{C}$ and with respect to the complex topology, ${\mathcal{M}}_{q,n}$ is a smooth vector bundle over ${\mathbb{P}}^{q-1}$. We prove that, in this case, ${\mathcal{M}}_{q,n}$ is diffeomorphic to a direct sum of twisted tangent bundles. We also prove that ${\mathcal{M}}_{q,n}$ possesses a universal property and represents a functor of ideals, and use it to identify ${\mathcal{M}}_{q,n}$ with an open subscheme of a punctual Hilbert scheme. By using a result of A. Iarrobino, we show that ${\mathcal{M}}_{q,n}\to {\mathbb{P}}^{q-1}$ is not a vector bundle, hence giving a family of affine space bundles that are not vector bundles.

17 pages