Mumford described a curve,
γ
\gamma
, in
P
3
{{\mathbf {P}}^3}
that has obstructed infinitesimal deformations (in fact the Hilbert scheme of the curve is generically nonreduced). This paper studies
γ
′
s
\gamma ’{\text {s}}
Hilbert scheme by studying deformations of
γ
\gamma
in
P
3
{{\mathbf {P}}^3}
over parameter spaces of the form
Spec
(
k
[
t
]
/
(
t
n
)
)
,
n
=
2
,
3
,
…
\operatorname {Spec} (k[t]/({t^n})),\,n = 2,\,3,\, \ldots
. Given a deformation of
γ
\gamma
over
Spec
(
k
[
t
]
/
(
t
n
)
)
\operatorname {Spec} (k[t]/({t^n}))
one attempts to extend it to a deformation of
γ
\gamma
over
Spec
(
k
[
t
]
/
(
t
n
+
1
)
)
\operatorname {Spec} (k[t]/({t^{n + 1}}))
. If it will not extend, this deformation is said to be obstructed at the nth order. I show that on a generic version of Mumford’s curve, an infinitesimal deformation (i.e., a deformation over
Spec
(
k
[
t
]
/
(
t
2
)
)
\operatorname {Spec} (k[t]/({t^2}))
) is either obstructed at the second order, or at no order, in which case we say it is unobstructed.