Picard Groups of Normal Surfaces

TL;DR

This paper investigates the singularities of general surfaces in P^3 containing a fixed subscheme Z, providing methods to classify singularities and compute the Picard group of these surfaces.

Contribution

It introduces a new algorithm to identify A_n singularities and offers a systematic approach to compute the Picard group of surfaces with prescribed base loci.

Findings

Classification of singularities at fixed points for general surfaces

Development of an algorithm to identify A_n singularities

Application of Noether-Lefschetz theory to compute Picard groups

Abstract

We study the fixed singularities imposed on members of a linear system of surfaces in P^3_C by its base locus Z. For a 1-dimensional subscheme Z \subset P^3 with finitely many points p_i of embedding dimension three and d >> 0, we determine the nature of the singularities p_i \in S for general S in |H^0 (P^3, I_Z (d))| and give a method to compute the kernel of the restriction map Cl S \to Cl O_{S,p_i}. One tool developed is an algorithm to identify the type of an A_n singularity via its local equation. We illustrate the method for representative Z and use Noether-Lefschetz theory to compute Pic S.