Obstructions to deforming curves on a prime Fano 3-fold

Abstract

We prove that for every smooth prime Fano $3$-fold $V$, the Hilbert scheme $\operatorname{Hilb}^{sc} V$ of smooth connected curves on $V$ contains a generically non-reduced irreducible component of Mumford type. We also study the deformations of degenerate curves $C$ in $V$, i.e., curves $C$ contained in a smooth anti-canonical member $S \in |-K_V|$ of $V$. We give a sufficient condition for $C$ to be stably degenerate, i.e., every small (and global) deformation of $C$ in $V$ is contained in a deformation of $S$ in $V$. As a result, by using the Hilbert-flag scheme of $V$, we determine the dimension and the smoothness of $\operatorname{Hilb}^{sc} V$ at the point $[C]$, assuming that the class of $C$ in $\operatorname{Pic} S$ is generated by $-K_V\big{\vert}_S$ together with the class of a line, or a conic on $V$.