K3 surfaces with 9 cusps in characteristic p

2019/02/05 by Toshiyuki Katsura, Matthias Schütt, Katsura, Toshiyuki +1
Arts and Humanities · Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Historical Studies and Socio-cultural Analysis

paper · pdf · doi:10.48550/arxiv.1902.01579

openalex publication_date 2019/02/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study K3 surfaces with 9 cusps, i.e. 9 disjoint A₂ configurations of smooth rational curves, over algebraically closed fields of characteristic p≠ 3. Much like in the complex situation studied by Barth, we prove that each such surface admits a triple covering by an abelian surface. Conversely, we determine which abelian surfaces with order three automorphisms give rise to K3 surfaces. We also investigate how K3 surfaces with 9 cusps hit the supersingular locus.

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