On the Classification of K3-Surfaces with Nine Cusps

1998/05/19 by W. Barth, Barth, W. · 3 citations
Computer Science · Mathematics · #14J15 #14J28 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #Digital Image Processing Techniques #FOS: Mathematics #math.AG #msc:14J15 #msc:14J28

paper · pdf · doi:10.48550/arxiv.math/9805082

arxiv created 1998/05/19 · openalex publication_date 1998/05/19 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

By a K3-surface with nine cusps I mean a compact complex surface with nine isolated double points A₂, but otherwise smooth, such that its minimal desingularisation is a K3-surface. In an earlier paper I showd that each such surface is a quotient of a complex torus by a cyclic group of order three. Here I try to classify these K3-surfaces, using the period map for complex tori. In particular I show: A K3-surface with nine cusps carries polarizations only of degrees 0 or 2 modulo 6. This implies in particular that there is no quartic surface in projective three-space with nine cusps. (T. Urabe pointed out to me how to deduce this from a theorem of Nikulin.) In an appendix I give explicit equations of quartic surfaces in three-space with eight cusps.

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