Picard groups of normal surfaces

John Brevik and Scott Nollet

Journal of Singularities
volume 4 (2012), 154-170

Received: 25 July 2012. Received in revised form: 14 October 2012.

DOI: 10.5427/jsing.2012.4i

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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 contained in 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 from 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.

Mathematical Subject Classification:

14B07, 14H10, 14H50

Author(s) information:

John Brevik Scott Nollet
California State University at Long Beach Texas Christian University
Department of Mathematics and Statistics Department of Mathematics
Long Beach, CA 90840 Fort Worth, TX 76129
email: jbrevik@csulb.edu email: s.nollet@tcu.edu