We consider the space Mq,n of regular q-tuples of commuting nilpotent endomorphisms of kn modulo simultaneous conjugation. We show that Mq,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, Mq,n is a smooth vector bundle over Pq−1. We prove that, in this case, Mq,n is diffeomorphic to a direct sum of twisted tangent bundles. We also prove that Mq,n possesses a universal property and represents a functor of ideals, and we use it to identify Mq,n with an open subscheme of a punctual Hilbert scheme. Using a result of A. Iarrobino’s, we show that Mq,n → Pq−1 is not a vector bundle, hence giving a family of affine space bundles that are not vector bundles.
ASJC Scopus subject areas
- Applied Mathematics