The classification of ACM curves on a surface in $\mathbb{P}^{3}$

TL;DR

This paper classifies arithmetically Cohen-Macaulay (ACM) curves on surfaces in projective 3-space, describes their geometric properties on quartic surfaces, and extends the classification to higher codimension subvarieties.

Contribution

It introduces a classification framework for ACM curves on surfaces using weak admissible pairs and generalizes the results to higher codimension subvarieties.

Findings

Classification of ACM curves via weak admissible pairs

Geometric description of ACM curves on smooth determinantal quartic surfaces

Computation of Picard classes for these curves

Abstract

We classify ACM curves contained in a surface of degree d in $\mathbb{P}^{3}$ in terms of weak admissible pairs. In the case of a very general smooth determinantal quartic surface, we provide a geometric description of these curves and compute their Picard classes on the surface. Finally, we present a generalization to ACM closed subvarieties of codimension $1$ on a hypersurface in $\mathbb{P}^{n}$.