Obstructions to deforming curves on a prime Fano 3-fold

TL;DR

This paper investigates the deformation theory of curves on smooth prime Fano 3-folds, revealing obstructions and conditions for stable degeneracy, and characterizes the local structure of the Hilbert scheme of such curves.

Contribution

It establishes the existence of non-reduced components in the Hilbert scheme of curves on Fano 3-folds and provides criteria for stable degeneracy of curves within these varieties.

Findings

Existence of non-reduced irreducible components in the Hilbert scheme.

Conditions under which curves are stably degenerate.

Determination of the Hilbert scheme's dimension and smoothness at specific points.

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 $V$.