Noether-Lefschetz locus and generalisation of an example due to Mumford
  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. 
Ananyo Dan
math.AG