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

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