2016/09/07 by Stephen Coughlan, Giancarlo Urzúa, Coughlan, Stephen +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1609.02177
openalex publication_date 2016/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that Godeaux--Reid surfaces with torsion group Z/3 have topological fundamental group Z/3. For this purpose, we describe degenerations to stable KSBA surfaces with one 1/4(1,1) singularity, whose minimal resolution are elliptic fibrations with two multiplicity 3 fibres and one I4 singular fibre. We study special such degenerations which have an involution, describing the corresponding Campedelli double plane construction. We also find some stable rational degenerations, some of which have more singularities, and one of which has a single 1/9(1,2) singularity, the minimal possible index for such a surface. Finally, we do the analogous study for the Godeaux surfaces with torsion Z/4.