Obstructions to Deforming Space Curves and Non-reduced Components of the Hilbert Scheme

Abstract

Let $H_{P^3}^S$ denote the Hilbert scheme of smooth connected curves in $P^3$. We consider maximal irreducible closed subsets $W\subset H_{P^3}^S$ whose general member $C$ is contained in a smooth cubic surface and investigate the conditions for $W$ to be a component of $(H_{P^3}^S{)}_{\mathrm{red}}$. We especially study the case where the dimension of the tangent space of $H_{P^3}^S$ at $[C]$ is greater than $\dim W$ ($\ge 4\mathrm{deg}(C)$) by one. We compute obstructions to deforming $C$ in $P^3$ and prove that for every $W$ in this case, $H_{P^3}^S$ is non-reduced along $W$ and $W$ is a component of $(H_{P^3}^S{)}_{\mathrm{red}}$.