Picard Groups of Normal Surfaces

الأصل · EN

We study the fixed singularities imposed on members of a linear system of surfaces in P³C by its base locus Z. For a 1-dimensional subscheme Z ⊂ P³ with finitely many points pᵢ of embedding dimension three and d >> 0, we determine the nature of the singularities pᵢ ∈ S for general S in |H⁰ (P³, IZ (d))| and give a method to compute the kernel of the restriction map Cl S → Cl Oₛ,ₚᵢ. One tool developed is an algorithm to identify the type of an Aₙ singularity via its local equation. We illustrate the method for representative Z and use Noether-Lefschetz theory to compute Pic S.