Conjugacy classes of commuting nilpotents

Abstract

We consider the space M q , n \mathcal {M}_{q,n} of regular q q -tuples of commuting nilpotent endomorphisms of k n k^n modulo simultaneous conjugation. We show that M q , n \mathcal {M}_{q,n} admits a natural homogeneous space structure, and that it is an affine space bundle over P q − 1 {\mathbb {P}}^{q-1} . A closer look at the homogeneous structure reveals that, over C {\mathbb {C}} and with respect to the complex*1pt topology, M q , n \mathcal {M}_{q,n} is a smooth vector bundle over P q − 1 {\mathbb {P}}^{q-1} . We prove that, in this case, M q , n \mathcal {M}_{q,n} is diffeomorphic to a direct sum of twisted tangent bundles. We also prove that M q , n \mathcal {M}_{q,n} possesses a universal property and represents a functor of ideals, and we use it to identify M q , n \mathcal {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 → P q − 1 \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.