On Castelnuovo theory and non-existence of smooth isolated curves in quintic threefolds

TL;DR

This paper establishes necessary conditions and non-existence results for smooth isolated curves in quintic threefolds, using Castelnuovo theory, and confirms the completeness of known examples up to degree 9.

Contribution

It introduces new non-existence criteria for smooth isolated curves in quintic threefolds and verifies the completeness of known examples up to degree 9.

Findings

Derived necessary conditions for isolated curves in quintic threefolds.

Proved non-existence of certain smooth curves in these threefolds.

Confirmed the completeness of Knutsen's list up to degree 9.

Abstract

We give some necessary conditions for a smooth irreducible curve $C\subset \mathbb{P}^4$ to be isolated in a smooth quintic threefold, and also find a lower bound for $h^1(\mathcal{N}_{C/{\mathbb{P}^4}})$. Combining these with beautiful results in Castelnuovo theory, we prove certain non-existence results on smooth curves in smooth quintic threefolds. As an application, we can prove Knutsen's list of examples of smooth isolated curves in general quintic threefolds is complete up to degree 9.