Beauville surfaces without real structures

Bauer, Ingrid; Catanese, Fabrizio; Grunewald, Fritz

doi:10.1007/0-8176-4417-2_1

Cited by 65 publications

(159 citation statements)

References 8 publications

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“…2) Let n ≥ 7 and let p be an odd prime such that n is not congruent to 0 or 1(mod p) (cf. 7.29), we get by the same arguments as in [BCG05a], prop. 5.5. that Σ(a, c) ∩ Σ(a ′ , c ′ ) = {1}.…”

Section: Theorem 719 the Following Groups Admit Unmixed Beauville Ssupporting

confidence: 58%

“…e.g. [Cat03], [BC04], [BCG05a]). Notice, that given a surface S = (C 1 × C 2 )/G isogenous to a product, we obtain always three more, exchanging C 1 with its conjugate curvē C 1 , or C 2 withC 2 , but only if we conjugate both C 1 and C 2 , we obtain an orientedly diffeomorphic surface.…”

Section: Corresponding To Surfaces Homeomorhphic Resp Diffeomorphmentioning

confidence: 99%

“…The Galois group acts transforming such a variety X to another variety of the same type, but whose fundamental group need not be isomorphic to the fundamental group of X. The last chapter is thus devoted to an overview of [BCG05a], where the so-called Beauville surfaces are studied. Beauville surfaces are rigid surfaces which admit a finite unramified covering which is a product of two algebraic curves.…”

Section: Introductionmentioning

confidence: 99%

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

Bauer

Catanese

Grunewald

2006

MedJM

Self Cite

161

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Abstract. We start discussing the group of automorphisms of the field of complex numbers, and describe, in the special case of polynomials with only two critical values, Grothendieck's program of 'Dessins d' enfants', aiming at giving representations of the absolute Galois group. We describe Chebycheff and Belyi polynomials, and other explicit examples. As an illustration, we briefly treat difference and Schur polynomials. Then we concentrate on a higher dimensional analogue of the triangle curves, namely, Beauville surfaces and varieties isogenous to a product. We describe their moduli spaces, and show how the study of these varieties leads to new interesting questions in the theory of finite (simple) groups.

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Section: Theorem 719 the Following Groups Admit Unmixed Beauville Ssupporting

confidence: 58%

Section: Corresponding To Surfaces Homeomorhphic Resp Diffeomorphmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Bauer

Catanese

Grunewald

2006

MedJM

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161

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“…The question was finally answered in the negative, and in every possible way ( [26,98,104,239,278]). The Friedman-Morgan speculation does not hold true and the DEF = DIFF question has a negative answer.…”

Section: Connected Components Of Gieseker's Moduli Spacementioning

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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“…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%

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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Beauville surfaces without real structures

Cited by 65 publications

References 8 publications

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

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

Topological methods in moduli theory

On Beauville surfaces

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