Obstructions to deforming space curves and non-reduced components of the Hilbert scheme

Abstract

Let $Hilb P^3$ denote the Hilbert scheme of smooth connected curves in $P^3$. We consider maximal irreducible closed subsets $W \subset Hilb P^3$ whose general member $C$ is contained in a smooth cubic surface and investigate the conditions for $W$ to be a component of $(Hilb P^3)_{red}$. We especially study the case where the dimension of the tangent space of $Hilb P^3$ at $[C]$ is greater than $\dim W$ by one. We compute obstructions to deforming $C$ in $P^3$ and prove that for every $W$ in this case, $Hilb P^3$ is non-reduced along $W$ and $W$ is a component of $(Hilb P^3)_{red}$.