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

TL;DR

This paper classifies non-ACM curves on smooth quintic hypersurfaces in projective three-space, providing insights into their properties beyond traditional invariants like degree and genus.

Contribution

It offers a novel classification of non-ACM curves on quintic hypersurfaces, expanding understanding of their geometric and algebraic characteristics.

Findings

Identifies criteria distinguishing aCM from non-aCM curves

Provides a comprehensive classification scheme for non-aCM curves

Enhances understanding of curve properties on quintic hypersurfaces

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.