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 Tr. Mat. Inst. Steklova, 2016, Volume 294, Pages 105–140 (Mi tm3736)

Plane rational quartics and K3 surfaces

Vik. S. Kulikov

Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia

Abstract: We study actions of the symmetric group $\mathbb S_4$ on K3 surfaces $X$ that satisfy the following condition: there exists an equivariant birational contraction $\overline r\colon X\to\overline X$ to a K3 surface $\overline X$ with ADE singularities such that the quotient space $\overline X/\mathbb S_4$ is isomorphic to $\mathbb P^2$. We prove that up to smooth equivariant deformations there exist exactly 15 such actions of the group $\mathbb S_4$ on K3 surfaces, and that these actions are realized as actions of the Galois groups on the Galoisations $\overline X$ of the dualizing coverings of the plane which are associated with plane rational quartics without $A_4$, $A_6$, and $E_6$ singularities as their singular points.

 Funding Agency Grant Number Russian Science Foundation 14-50-00005 This work is supported by the Russian Science Foundation under grant 14-50-00005.

DOI: https://doi.org/10.1134/S0371968516030079

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English version:
Proceedings of the Steklov Institute of Mathematics, 2016, 294, 95–128

Bibliographic databases:

UDC: 512.77

Citation: Vik. S. Kulikov, “Plane rational quartics and K3 surfaces”, Modern problems of mathematics, mechanics, and mathematical physics. II, Collected papers, Tr. Mat. Inst. Steklova, 294, MAIK Nauka/Interperiodica, Moscow, 2016, 105–140; Proc. Steklov Inst. Math., 294 (2016), 95–128

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tm3736
• https://doi.org/10.1134/S0371968516030079
• http://mi.mathnet.ru/eng/tm/v294/p105

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This publication is cited in the following articles:
1. Vik. S. Kulikov, “The Hesse curve of a Lefschetz pencil of plane curves”, Russian Math. Surveys, 72:3 (2017), 574–576
2. V. V. Nikulin, “Degenerations of Kählerian K3 surfaces with finite symplectic automorphism groups. III”, Izv. Math., 81:5 (2017), 985–1029
3. Vik. S. Kulikov, E. I. Shustin, “On $G$-Rigid Surfaces”, Proc. Steklov Inst. Math., 298 (2017), 133–151
4. Vik. S. Kulikov, “On divisors of small canonical degree on Godeaux surfaces”, Sb. Math., 209:8 (2018), 1155–1163
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