Noether-Lefschetz theorem with base locus

TL;DR

This paper extends the Noether-Lefschetz theorem to normal surfaces with arbitrary base loci in complex projective three-space, providing new methods for computing class groups and Picard groups.

Contribution

It introduces a new approach to compute class groups of surfaces with base loci, generalizing the classic theorem and simplifying existing proofs.

Findings

Computed class groups for surfaces with base loci

Generalized Picard group calculations

Extended Noether-Lefschetz theorem to new cases

Abstract

We compute the class groups of very general normal surfaces in complex projective three-space containing an arbitrary base locus $Z$, thereby extending the classic Noether-Lefschetz theorem (the case when $Z$ is empty). Our method is an adaptation of Griffiths and Harris' degeneration proof, simplified by a cohomology and base change argument. We give applications to computing Picard groups, which generalize several known results.