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2013 | OriginalPaper | Chapter

Enriques Surfaces of Hutchinson–Göpel Type and Mathieu Automorphisms

Authors : Shigeru Mukai, Hisanori Ohashi

Published in: Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds

Publisher: Springer New York

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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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previous chapter A Note About Special Cycles on Moduli Spaces of K3 Surfaces

next chapter Quartic K3 Surfaces and Cremona Transformations

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.

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].

In fact only g = 2 is possible.

This means that every involution is Mathieu.

An automorphism is semi-symplectic if it acts on the space \({H}^{0}(S,\mathcal{O}_{S}(2K_{S}))\) trivially.

W. Barth, K. Hulek, C. Peters, A. Van de Ven, in Compact Complex Surfaces, 2nd enlarged edn. Erg. der Math. und ihrer Grenzgebiete, 3. Folge, Band 4 (Springer, Berlin, 2004)

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Title: Enriques Surfaces of Hutchinson–Göpel Type and Mathieu Automorphisms
Authors: Shigeru Mukai
Hisanori Ohashi
Publisher: Springer New York
Book: Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds
Print ISBN: 978-1-4614-6402-0

Electronic ISBN: 978-1-4614-6403-7

Copyright Year: 2013
DOI: https://doi.org/10.1007/978-1-4614-6403-7_15

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