2005/01/31 by Alberto Calabri, Ciro Ciliberto, Calabri, Alberto +3 · 1 citation
Engineering · Mathematics · #14J29 #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #math.AG #msc:14J29
paper · pdf · doi:10.48550/arxiv.math/0501551
25 pages, 1 figure (with Xy-pic)
arxiv created 2005/01/31 · openalex publication_date 2005/01/31 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Minimal algebraic surfaces of general type with the smallest possible invariants have geometric genus zero and K2=1 and are usually called "numerical Godeaux surfaces". Although they have been studied by several authors, their complete classification is not known. In this paper we classify numerical Godeaux surfaces with an involution, i.e. with an automorphism of order 2. We prove that they are birationally equivalent either to double covers of Enriques surfaces, or to double planes of two different types: the branch curve either has degree 10 and suitable singularities, originally suggested by Campedelli, or is the union of two lines and a curve of degree 12 with certain singularities. The latter type of double planes are degenerations of examples described by Du Val and their existence was previously unknown; we show some examples of this new type, computing also their torsion group.