Obstructions to deforming curves on an Enriques-Fano 3-fold

Abstract

We study the deformations of a curve $C$ on an Enriques-Fano $3$-fold $X\subset {\mathbb{P}}^n$, assuming that $C$ is contained in a smooth hyperplane section $S\subset X$, that is a smooth Enriques surface in $X$. We give a sufficient condition for $C$ to be (un)obstructed in $X$, in terms of half pencils and $(-2)$-curves on $S$. Let ${\mathrm{Hilb}}^{sc}X$ denote the Hilbert scheme of smooth connected curves in $X$. By using the Hilbert-flag scheme of $X$, we also compute the dimension of ${\mathrm{Hilb}}^{sc}X$ at $[C]$ and give a sufficient condition for ${\mathrm{Hilb}}^{sc}X$ to contain a generically non-reduced irreducible component of Mumford type.

18 pages, final version, to appear in Journal of Pure and Applied Algebra