Constructions of Kummer structures on generalized Kummer surfaces

2021/09/05 by Xavier Roulleau, A. Sarti, Roulleau, Xavier +1
Mathematics · #14J28 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.2109.02146

openalex publication_date 2021/09/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study generalized Kummer surfaces Km₃(A), by which we mean the K3 surfaces obtained by desingularization of the quotient of an abelian surface A by an order 3 symplectic automorphism group. Such a surface carries 9 disjoint configurations of two smooth rational curves C,C' with CC'=1. This 9\bf A₂-configuration plays a role similar to the Nikulin configuration of 16 disjoint smooth rational curves on (classical) Kummer surfaces. We study the (generalized) question of T. Shioda: suppose that Km₃(A) is isomorphic to Km₃(B), does that imply that A and B are isomorphic? We answer by the negative in general, by two methods: by a link between that problem and Fourier-Mukai partners of A, and by construction of 9\bf A₂-configurations on Km₃(A) which cannot be exchanged under the automorphism group.

Citations

• A refinement of Bézout's Lemma, and order 3 elements in some quaternion algebras over ℚ
• On the Classification of K3-Surfaces with Nine Cusps

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• Explicit Nikulin configurations on Kummer surfaces
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