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

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 ${\mathbb{P}}^n$.