Conjugacy classes of commuting nilpotents

TL;DR

This paper studies the geometric structure of the space of commuting nilpotent endomorphisms, revealing it as an affine space bundle over projective space with a universal property and connections to Hilbert schemes.

Contribution

It characterizes the space of commuting nilpotent endomorphisms as an affine space bundle, identifies its smooth vector bundle structure over complex numbers, and relates it to Hilbert schemes.

Findings

$ ext{M}_{q,n}$ is a homogeneous space and an affine space bundle over $ ext{P}^{q-1}$.

Over $ ext{C}$, $ ext{M}_{q,n}$ is a smooth vector bundle and a direct sum of twisted tangent bundles.

$ ext{M}_{q,n}$ has a universal property and is an open subscheme of a punctual Hilbert scheme.

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.