On the classification of non-aCM curves on quintic hypersurfaces in ${\mathbb{P}}^3$

Abstract

In this paper, we call a sub-scheme of dimension one in ${\mathbb{P}}^3$ a curve. It is well known that the arithmetic genus and the degree of an aCM curve $D$ in ${\mathbb{P}}^3$ is computed by the $h$-vector of $D$. However, for a given curve $D$ in ${\mathbb{P}}^3$, the two invariants of $D$ do not tell us whether $D$ is aCM or not. In this paper, we give a classification of curves on a smooth quintic hypersurface in ${\mathbb{P}}^3$ which are not aCM.