Irreducibility of the space of dihedral covers of the projective line of a given numerical type

Catanese, Fabrizio; Lönne, Michael; Perroni, Fabio

doi:10.4171/rlm/601

Cited by 31 publications

(50 citation statements)

References 18 publications

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“…The above result completes the classification of the unmarked topological types for G = D n , begun in [112]; moreover this result entails the classification of the irreducible components of the loci M g,D n (see the appendix to [115]). …”

Section: Remark 207mentioning

confidence: 77%

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: Remark 207mentioning

confidence: 77%

Topological methods in moduli theory

Catanese

2015

Bull. Math. Sci.

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“…So the submanifold T(G, θ) is not well-defined, but the subvariety M(G, θ) is well-defined. For more details see [46,8,6].…”

Section: Special Subvarieties In the Unramified Prym Locusmentioning

confidence: 99%

Shimura curves in the Prym locus

Colombo

Frediani

Ghigi

et al. 2019

Commun. Contemp. Math.

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Abstract. We study Shimura curves of PEL type in Ag generically contained in the Prym locus. We study both the unramified Prym locus, obtained usingétale double covers, and the ramified Prym locus, corresponding to double covers ramified at two points. In both cases we consider the family of all double covers compatible with a fixed group action on the base curve. We restrict to the case where the family is 1-dimensional and the quotient of the base curve by the group is P 1 . We give a simple criterion for the image of these families under the Prym map to be a Shimura curve. Using computer algebra we check all the examples gotten in this way up to genus 28. We obtain 44 Shimura curves generically contained in the unramified Prym locus and 9 families generically contained in the ramified Prym locus. Most of these curves are not generically contained in the Jacobian locus.

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“…It is a complex submanifold of dimension 3g ′ − 3+ r, isomorphic to the Teichmüller space T g ′ ,r (see e.g. [3,14]). This isomorphism can be described as follows: if (C, ϕ) is a curve with a marking such that…”

Section: In Fact Letmentioning

confidence: 99%

“…All the other critical points of ψ have stabilisers of order 2, hence they are not critical values for the map ϕ. So the map ϕ has ramification (3,3) and by the unicity argument 4.2 we can assume that ϕ : X → X/G belongs to our family (3). Concluding, the special subvariety given by family (3) gives the same special subvariety obtained as the family (31) of Galois coverings of P 1 via S 3 found in [13].…”

Section: Example (3)mentioning

confidence: 99%

Shimura varieties in the Torelli locus via Galois coverings of elliptic curves

2015

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Abstract. We study Shimura subvarieties of Ag obtained from families of Galois coverings f : C → C ′ where C ′ is a smooth complex projective curve of genus g ′ ≥ 1 and g = g(C). We give the complete list of all such families that satisfy a simple sufficient condition that ensures that the closure of the image of the family via the Torelli map yields a Shimura subvariety of Ag for g ′ = 1, 2 and for all g ≥ 2, 4 and for g ′ > 2 and g ≤ 9. In [13] similar computations were done in the case g ′ = 0. Here we find 6 families of Galois coverings, all with g ′ = 1 and g = 2, 3, 4 and we show that these are the only families with g ′ = 1 satisfying this sufficient condition. We show that among these examples two families yield new Shimura subvarieties of Ag, while the other examples arise from certain Shimura subvarieties of Ag already obtained as families of Galois coverings of P 1 in [13]. Finally we prove that if a family satisfies this sufficient condition with g ′ ≥ 1, then g ≤ 6g ′ + 1.

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Irreducibility of the space of dihedral covers of the projective line of a given numerical type

Cited by 31 publications

References 18 publications

Topological methods in moduli theory

Topological methods in moduli theory

Shimura curves in the Prym locus

Shimura varieties in the Torelli locus via Galois coverings of elliptic curves

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