Abstract
We study a class of Enriques surfaces called of Hutchinson–Göpel type. Starting with the projective geometry of Jacobian Kummer surfaces, we present the Enriques’ sextic expression of these surfaces and their intrinsic symmetry by \(G = C_{2}^{3}\). We show that this G is of Mathieu type and conversely, that these surfaces are characterized among Enriques surfaces by the group action by \(C_{2}^{3}\) with prescribed topological type of fixed point loci. As an application, we construct Mathieu type actions by the groups \(C_{2} \times \mathfrak{A}_{4}\) and \(C_{2} \times C_{4}\). Two introductory sections are also included.
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Notes
- 1.
This octic model of \(\mathrm{Km}\,C\) is different from the standard nonsingular octic model given by the smooth complete intersection of three diagonal quadrics. See (⋆2) of Sect. 5.
- 2.
This number is exactly the number of fixed points of non-free involutions in the small Mathieu group M 12, which implies that the character of Mathieu involutions on \({H}^{{\ast}}(S, \mathbb{Q})\) coincides with that of involutions in M 11. This is the origin of the terminology. See also [11].
- 3.
In fact only g = 2 is possible.
- 4.
This means that every involution is Mathieu.
- 5.
An automorphism is semi-symplectic if it acts on the space \({H}^{0}(S,\mathcal{O}_{S}(2K_{S}))\) trivially.
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Acknowledgements
We are grateful to the organizers of the interesting Workshop on Arithmetic and Geometry of K3 surfaces and Calabi–Yau threefolds. The second author is grateful to Professor Shigeyuki Kondo for discussions and encouragement.
This work is supported in part by the JSPS Grant-in-Aid for Scientific Research (B) 17340006, (S) 19104001, (S) 22224001, (A) 22244003, for Exploratory Research 20654004 and for Young Scientists (B) 23740010.
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Mukai, S., Ohashi, H. (2013). Enriques Surfaces of Hutchinson–Göpel Type and Mathieu Automorphisms. In: Laza, R., Schütt, M., Yui, N. (eds) Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds. Fields Institute Communications, vol 67. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6403-7_15
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DOI: https://doi.org/10.1007/978-1-4614-6403-7_15
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