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

## TL;DR

This paper investigates the deformation theory of curves on Enriques-Fano 3-folds, providing conditions for obstructions and analyzing the structure of the Hilbert scheme of such curves.

## Contribution

It offers new criteria for the (un)obstructedness of curves on Enriques-Fano 3-folds and characterizes components of the Hilbert scheme with non-reduced structure.

## Key findings

- Provides sufficient conditions for curve deformations to be obstructed or unobstructed.
- Calculates the dimension of the Hilbert scheme at specific points.
- Identifies conditions for the Hilbert scheme to have non-reduced components.

## 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 $\operatorname{Hilb}^{sc} X$ denote the Hilbert scheme of smooth connected curves in $X$. By using the Hilbert-flag scheme of $X$, we also compute the dimension of $\operatorname{Hilb}^{sc} X$ at $[C]$ and give a sufficient condition for $\operatorname{Hilb}^{sc} X$ to contain a generically non-reduced irreducible component of Mumford type.