Chebycheff and Belyi Polynomials, Dessins d’Enfants, Beauville Surfaces and Group Theory

Bauer, Ingrid; Catanese, Fabrizio; Grunewald, Fritz

doi:10.1007/s00009-006-0069-7

Cited by 47 publications

(163 citation statements)

References 23 publications

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“…Since the argument for the above theorem is not constructive, let us observe that, in work in collaboration with Bauer and Grunewald [27,29], we discovered wide classes of explicit algebraic surfaces isogenous to a product for which the same phenomenon holds.…”

Section: Theorem 230 If σ ∈ Gal(q/q) Is Not In the Conjugacy Class Ofmentioning

confidence: 99%

Topological methods in moduli theory

Catanese

2015

Bull. Math. Sci.

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One of the main themes of this long article is the study of projective varieties which are K(H,1)'s, i.e. classifying spaces BH for some discrete group H. After recalling the basic properties of such classifying spaces, an important class of such varieties is introduced, the one of Bagnera-de Franchis varieties, the quotients of an Abelian variety by the free action of a cyclic group. Moduli spaces of Abelian varieties and of algebraic curves enter into the picture as examples of rational K(H,1)'s, through Teichmüller theory. The main trhust of the paper is to show how in the case of K(H,1)'s the study of moduli spaces and deformation classes can be achieved through by now classical results concerning regularity of classifying maps. The Inoue type varieties of Bauer and Catanese are introduced and studied as a key example, and new results are shown. Motivated from this study, the moduli spaces of algebraic varieties, and especially of algebraic curves with a group of automorphisms of a given topological type are studied in detail, following new results by the author, Michael Lönne and Fabio Perroni. Finally, the action of the absolute Galois group on the moduli spaces of such K(H,1) varieties is studied. In the case of surfaces isogenous to a product, it is shown how this yields a faifhtul action on the set of connected components of the moduli space: for each Galois automorphism of order different from 2 there is an Communicated by Efim Zelmanov.The present work took place in the realm of the DFG Forschergruppe 790 "Classification of algebraic surfaces and compact complex manifolds". Part of the article was written when the author was visiting KIAS, Seoul, as KIAS research scholar.

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Section: Theorem 230 If σ ∈ Gal(q/q) Is Not In the Conjugacy Class Ofmentioning

confidence: 99%

Topological methods in moduli theory

Catanese

2015

Bull. Math. Sci.

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“…Among the complex surfaces defined over a number field an important class is that of Beauville surfaces defined as follows Definition ( [3]). A Beauville surface is a compact complex surface S satisfying the following properties:…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…Most, if not all, of what is known about them is due to work done by Catanese on his own ( [5]) or jointly with Bauer and Grunewald ( [3], [2]). …”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…As noted in [3] there are many interesting open problems regarding Beauville surfaces. The most obvious ones are questions such as which finite groups can occur, which curves C i , which genera g i , which curves or genera for a given group G, etc.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…As shown in [2] and [3] there is a purely algebraic criterion to detect when a finite group G admits a Beauville structure. Definition 1.…”

Section: A Criterion For G To Admit Beauville Structuresmentioning

confidence: 99%

See 2 more Smart Citations

On Beauville surfaces

Fuertes¹,

González-Diez²,

Jaikin–Zapirain³

2011

Groups Geom. Dyn.

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Abstract. We prove that if a finite group G acts freely on a product of two curves C 1 C 2 so that the quotient S D C 1 C 2 =G is a Beauville surface then C 1 and C 2 are both non hyperelliptic curves of genus 6; the lowest bound being achieved when C 1 D C 2 is the Fermat curve of genus 6 and G D .Z=5Z/ 2 . We also determine the possible values of the genera of C 1 and C 2 when G equals S 5 , PSL 2 .F 7 / or any abelian group. Finally, we produce examples of Beauville surfaces in which G is a p-group with p D 2; 3.Mathematics Subject Classification (2010). 14J29, 20D06, 30F10.

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Strongly Real Beauville Groups III

Fairbairn

2020

Springer Proceedings in Mathematics &Amp; Statistics

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Beauville surfaces are a class of complex surfaces defined by letting a finite group G act on a product of Riemann surfaces. These surfaces possess many attractive geometric properties several of which are dictated by properties of the group G. A particularly interesting subclass are the 'strongly real' Beauville surfaces that have an analogue of complex conjugation defined on them. In this survey we discuss these objects and in particular the groups that may be used to define them. En route we discuss several open problems, questions and conjectures and in places make some progress made on addressing these.

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Chebycheff and Belyi Polynomials, Dessins d’Enfants, Beauville Surfaces and Group Theory

Cited by 47 publications

References 23 publications

Topological methods in moduli theory

Topological methods in moduli theory

On Beauville surfaces

Strongly Real Beauville Groups III

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