  In this article we generalise the well-known example due to Mumford for a
generically non-reduced component of the Hilbert scheme of curves in
$\mathbb{P}^3$ whose general element is smooth. The example considers smooth
curves in smooth cubic surfaces in $\mathbb{P}^3$. In this article we give
similar examples of generically non-reduced component of the Hilbert scheme of
curves in $\mathbb{P}^3$, for every integer $d \ge 5$, whose general element is
a smooth curve contained in a smooth degree $d$ surface in $\mathbb{P}^3$ and
not in any surface of smaller degree. The techniques used are motivated by the
study of Noether-Lefschetz locus.
