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

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 $h^1({\mathcal{N}}_{C\mathrm{/}{\mathbb{P}}^4})$. 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.