Some Remarks on the Wiman–Edge Pencil

Zamora, Alexis G.

doi:10.1017/s0013091517000232

Cited by 7 publications

(3 citation statements)

References 12 publications

(19 reference statements)

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“…The only known example of a fibration with exactly 5 singular fibers and non hyperelliptic general fiber seems to be the one induced by the Wiman-Edge pencil. In this example, f is the resolution of a plane pencil of degree 6 and genus 6 with base points not in a general position (see, for instance [8], [9], [20]). In this way, this example shows that there exists at least one fibration satisfying the hypothesis of Theorem 1.1 iv).…”

Section: The Conclusion Of the Proof Of Theorem 11mentioning

confidence: 99%

Semistable fibrations over $$\mathbb {P}^1$$ P 1 with five singular fibers

Castañeda-Salazar

Zamora

2017

Bol. Soc. Mat. Mex.

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Given a semistable non isotrivial fibration f : X → P 1 with general non hyperelliptic fiber of genus g ≥ 4, we show that if the number s of singular fibers is 5, then the genus is ≤ 11, thus improving the previously known bound g ≤ 17. Furthermore we show that for each possible genus the general fiber has gonality at most 5 and we describe the corresponding fibrations as the resolution of concrete pencils of curves on minimal rational surfaces.

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Section: The Conclusion Of the Proof Of Theorem 11mentioning

confidence: 99%

Semistable fibrations over $$\mathbb {P}^1$$ P 1 with five singular fibers

Castañeda-Salazar

Zamora

2017

Bol. Soc. Mat. Mex.

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“…For a number of recent papers on the Wiman-Edge pencil, see [3,4,5,10]. Given a family of varieties, it is a basic problem to compute its monodromy, to relate this to geometric properties of the family, and to use this information to uniformize (if possible) the base in terms of a period mapping, via Hodge structures.…”

Section: Introductionmentioning

confidence: 99%

Arithmeticity of the monodromy of the Wiman–Edge pencil

Farb¹,

Looijenga²

2022

Annales De l'Institut Fourier

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The Wiman-Edge pencil is the universal family of projective, genus 6, complex-algebraic curves endowed with a faithful action of the icosahedral group. The goal of this paper is to prove that its monodromy group is commensurable with a Hilbert modular group; in particular is arithmetic. We then give a modular interpretation of this, as well as a uniformization of its base.Résumé. -Le pinceau de Wiman-Edge est une famille universelle de courbes projectives non singulières de genre 6 et munie d'une action fidèle du groupe icosahédral. Le but principal de ce travail est la détermination de son groupe de monodromie. Nous montrons que ce groupe est arithmétique et commensurable avec un groupe modulaire de Hilbert. Nous donnons une interprétation modulaire de ce fait et décrivons en plus une uniformisation de la base.

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“…The base B of the Wiman pencil appears also as the moduli space of K3 surfaces with (a certain) faithful µ 2 × A 5 action; see §5.3 of [7]. For a number of recent papers on the Wiman-Edge pencil, see [2,3,5,7,12]. The problem of finding uniformizations of moduli spaces is a classical one, but it is typically a difficult task.…”

Section: Introductionmentioning

confidence: 99%

Geometry of the Wiman–Edge pencil and the Wiman curve

Farb

Looijenga

2020

Geom Dedicata

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The Wiman-Edge pencil is the universal family C t , t ∈ B of projective, genus 6, complex-algebraic curves admitting a faithful action of the icosahedral group A 5 . The curve C 0 , discovered by Wiman in 1895 [11] and called the Wiman curve, is the unique smooth, genus 6 curve admitting a faithful action of the symmetric group S 5 . In this paper we give an explicit uniformization of B as a non-congruence quotient Γ\H of the hyperbolic plane H, where Γ < PSL 2 (Z) is a subgroup of index 18. We also give modular interpretations for various aspects of this uniformization, for example for the degenerations of C t into 10 lines (resp. 5 conics) whose intersection graph is the Petersen graph (resp. K 5 ).In the second half of this paper we give an explicit arithmetic uniformization of the Wiman curve C 0 itself as the quotient Λ\H, where Λ is a principal level 5 subgroup of a certain "unit spinor norm" group of Möbius transformations. We then prove that C 0 is a certain moduli space of Hodge structures, endowing it with the structure of a Shimura curve of indefinite quaternionic type.

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Some Remarks on the Wiman–Edge Pencil

Cited by 7 publications

References 12 publications

Semistable fibrations over $$\mathbb {P}^1$$ P 1 with five singular fibers

Semistable fibrations over $$\mathbb {P}^1$$ P 1 with five singular fibers

Arithmeticity of the monodromy of the Wiman–Edge pencil

Geometry of the Wiman–Edge pencil and the Wiman curve

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