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2018 K3 surfaces with $\mathbb Z_2^2$ symplectic action
Luca Schaffler
Rocky Mountain J. Math. 48(7): 2347-2383 (2018). DOI: 10.1216/RMJ-2018-48-7-2347

Abstract

Let $G$ be a finite abelian group which acts symplectically on a K3 surface. The Néron-Severi lattice of the projective K3 surfaces admitting $G$ symplectic action and with minimal Picard number was computed by Garbagnati and Sarti. We consider a four-dimensional family of projective K3 surfaces with $\mathbb {Z}_2^2$ symplectic action which do not fall into the above cases. If $X$ is one of these K3 surfaces, then it arises as the minimal resolution of a specific $\mathbb {Z}_2^3$-cover of $\mathbb {P}^2$ branched along six general lines. We show that the Néron-Severi lattice of $X$ with minimal Picard number is generated by $24$ smooth rational curves and that $X$ specializes to the Kummer surface $\textrm{Km}(E_i\times E_i)$. We relate $X$ to the K3 surfaces given by the minimal resolution of the $\mathbb {Z}_2$-cover of $\mathbb {P}^2$, branched along six general lines, and the corresponding Hirzebruch-Kummer covering of exponent $2$ of $\mathbb {P}^2$.

Citation

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Luca Schaffler. "K3 surfaces with $\mathbb Z_2^2$ symplectic action." Rocky Mountain J. Math. 48 (7) 2347 - 2383, 2018. https://doi.org/10.1216/RMJ-2018-48-7-2347

Information

Published: 2018
First available in Project Euclid: 14 December 2018

zbMATH: 06999266
MathSciNet: MR3892136
Digital Object Identifier: 10.1216/RMJ-2018-48-7-2347

Subjects:
Primary: 14C22, 14J28, 14J50

Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium

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Vol.48 • No. 7 • 2018
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