  In this article, we study the Hilbert scheme of generically non-reduced
curves in $\mathbb{P}^3$. We prove the existence of generically non-reduced
curves in $\mathbb{P}^3$ for which there exist infinitesimal deformations of
the curve that do not induce deformations of the associated reduced scheme. We
show that such infinitesimal deformations contribute to the non-reducedness of
the corresponding Hilbert scheme. We introduce the notion of extension of
curves and prove that such infinitesimal deformations (such that the associated
reduced scheme does not deform) are inherited by the extended curve. Finally,
we give examples of extension of curves.
