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.