  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.
