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*1pt 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 we use it to identify $\mathcal {M}_{q,n}$ with an open subscheme of a punctual Hilbert scheme. Using a result of A. Iarrobino’s, 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.