Divisors class groups of singular surfaces

TL;DR

This paper computes the divisor class groups of singular surfaces, providing a sequence relating Cartier and almost Cartier divisors to those of the normalization, with applications to curves in P^3.

Contribution

It generalizes Hartshorne's theorem for cubic ruled surfaces and applies the results to classify certain curves in projective space.

Findings

Established an exact sequence relating divisors of singular surfaces and their normalization

Generalized Hartshorne's theorem to broader classes of surfaces

Limited the set of possible curves that are set-theoretic complete intersections in P^3

Abstract

We compute divisors class groups of singular surfaces. Most notably we produce an exact sequence that relates the Cartier divisors and almost Cartier divisors of a surface to the those of its normalization. This generalizes Hartshorne's theorem for the cubic ruled surface in P^3. We apply these results to limit the possible curves that can be set-theoretic complete intersection in P^3 in characteristic zero.