The classification of ACM curves on a surface in $\mathbb{P}^{3}$
Abel Castorena, Montserrat Vite
TL;DR
The work addresses the problem of classifying arithmetically Cohen–Macaulay curves on a degree $d$ surface $X\subset \mathbb{P}^3$ by introducing weak determinantal surfaces and weak admissible pairs. It proves a main classification: if $X$ is not weak determinantal, every ACM curve on $X$ is a complete intersection; if $X$ is weak determinantal of a type $(\hat{a},\hat{b})$, then non–complete-intersection ACM curves have Hilbert–Burch–type minimal free resolutions dictated by that type. The paper provides a complete treatment for degree $4$ surfaces, including twenty infinite families of weak admissible pairs and a geometric description for very general determinantal quartic surfaces, and it culminates with a higher-dimensional generalization to ACM codimension-one subvarieties on hypersurfaces. This work connects ACM curves, liaison theory, and determinantal geometry, offering explicit presentations and enabling deeper understanding of Noether–Lefschetz phenomena in quartic surfaces. The results yield concrete computational tools for resolving ACM curves on determinantal and non-determinantal surfaces and extend to higher dimensions via a uniform framework.
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 $\mathbb{P}^{n}$.