The Fundamental Group and Torsion Group of Beauville Surfaces

Bauer, Ingrid; Catanese, Fabrizio; Frapporti, Davide

doi:10.1007/978-3-319-13862-6_1

Cited by 11 publications

(22 citation statements)

References 27 publications

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“…Bagnera-de Franchis varieties and applications to moduli. In our article [BCF15b], appeared in the volume dedicated to Kodaira, we treated a special case of the theory of Inoue type varieties, the one where the action of G happens to be free also on Z, and Z is the simplest projective classifying space, an Abelian variety.…”

Section: Topological Methods For Modulimentioning

confidence: 99%

Kodaira fibrations and beyond: methods for moduli theory

Catanese¹

2017

Jpn. J. Math.

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Abstract. Kodaira fibred surfaces are a remarkable example of projective classifying spaces, and there are still many intriguing open questions concerning them, especially the slope question. The topological characterization of Kodaira fibrations is emblematic of the use of topological methods in the study of moduli spaces of surfaces and higher dimensional complex algebraic varieties, and their compactifications. Our tour through algebraic surfaces and their moduli (with results valid also for higher dimensional varieties) deals with fibrations, questions on monodromy and factorizations in the mapping class group, old and new results on Variation of Hodge Structures, especially a recent answer given (in joint work with Dettweiler) to a long standing question posed by Fujita. In the landscape of our tour, Galois coverings, deformations and rigid manifolds (there are by the way rigid Kodaira fibrations), projective classifying spaces, the action of the absolute Galois group on moduli spaces, stand also in the forefront. These questions lead to interesting algebraic surfaces, for instance remarkable surfaces constructed from VHS, surfaces isogenous to a product with automorphisms acting trivially on cohomology, hypersurfaces in Bagnera-de Franchis varieties, Inoue-type surfaces.

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Section: Topological Methods For Modulimentioning

confidence: 99%

Kodaira fibrations and beyond: methods for moduli theory

Catanese¹

2017

Jpn. J. Math.

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“…The unmixed ramification structure T = (T 1 , T 2 ) corresponds to S = (C × D)/G is described in [4]. They computed that T = (T 1 , T 2 ) is of type ([2, 2, 2, 4], [2,2,4,4]) on G(32, 27) which is equivalent to…”

Section: Equivariant Geometry Of C and Dmentioning

confidence: 99%

“…There is a semiorthogonal decomposition D b (S) = A, B where B is the full triangulated subcategory of D b (S) generated by exceptional collection of length 4 constructed above. From [21], we see that the Hochschild homology of A vanishes and from [2], we see that the Grothendieck group of A is Z 2 2 ⊕Z 4 ⊕Z 8 . Therefore A is a quasiphantom category.…”

Section: Quasiphantom Categoriesmentioning

confidence: 99%

Quasiphantom categories on a family of surfaces isogenous to a higher product

Kim

Lee

2017

Journal of Algebra

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We construct exceptional collections of line bundles of maximal length 4 on S = (C ×D)/G which is a surface isogenous to a higher product with p g = q = 0 where G = G(32, 27) is a finite group of order 32 having number 27 in the list of Magma library. From these exceptional collections, we obtain new examples of quasiphantom categories as their orthogonal complements.A surface S which is isomorphic to (C ×D)/G where C, D are curves of genus ≥ 2 and G is a finite group acting on C × D freely is called a surface isogenous to a higher product. Surfaces isogenous to a higher product are interesting and important classes of surfaces of general type. They play an important role in the study of moduli spaces of general type surfaces. Bauer, Catanese and Grunewald classified surfaces isogenous to a higher product of unmixed type with p g = q = 0 in [4]. In particular, they proved that the possible list ofA 5 cases. It is very natural to expect that derived categories of all surfaces isogenous to a higher product with p g = q = 0 have quasiphantom categories in their derived categories. However quasiphantom categories in derived categories of surfaces had not been constructed for G = G(32, 27), A 5 cases.In this paper we construct exceptional collections of line bundles of maximal length 4 on S = (C × D)/G which is a surface isogenous to a higher product with p g = q = 0 and G is G(32, 27). From these exceptional collections we can obtain new examples of quasiphantom categories. They are obtained as the orthogonal complements of the exceptional collections of maximal length 4 on S.Acknowledgement. We are grateful to Changho Keem, Young-Hoon Kiem for their words of encouragement. We thank Fabrizio Catanese for answering many questions and helpful discussions. We also thank Alexander Kuznetsov for his suggestion to use a computer program to compute the derived categories of the family of surfaces considered in this paper. Part of this work was done when the third named author was a research fellow of Korea Institute for Advanced Study. He thanks KIAS for wonderful working condition and kind hospitality.This work was also supported by IBS-R003-Y1. Last but not least, we thank the referees for their valuable comments which helped us to improve the manuscript.Notations. We will work over C. Derived category of a variety will mean the

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“…It is, however, somewhat limited in its usefulness and topologists have found several important higher dimensional analogues of the fundamental group and so it is natural to pose the following question. By way of partial progress on this question in [8] Bauer, Catanese and Frapporti recently showed that for any Beauville surface S the homology group H 1 (S, Z) is finite. They also give a much more detailed discussion of geometric aspects of the study of fundamental groups of Beauville surfaces and related objects as well as computer calculations of these objects in some cases.…”

Section: Fundamental Groups Of Beauville Surfacesmentioning

confidence: 99%