Geometry of the Winger pencil

Zi, Yunpeng

doi:10.1007/s40879-021-00485-6

“…Let U 0 ⊂ B be an open disk centered at s = 0 of radius < 27 5 . We proved in [13] that by doing a base change over U 0 of order 3 (with Galois group µ 3 ), given by t ∈ Û0 → t = t3 ∈ U 0 , the pull back of W U 0 /U 0 can be modified over the central fiber C 0 only to make it a smooth family Ŵ Û0 / Û0 which still retains the µ 3 -action. The central fiber is then a smooth curve Ĉ0 with an action of I × µ 3 whose µ 3 -orbit space gives K. This implies that the monodromy of the original family around 0 (which is a priori only given as an isotopy class of diffeomorphisms of a nearby smooth fiber) can be represented by the action of a generator φ ∈ µ 3 on Ĉ0 (which indeed commutes with the I-action on C 0 ).…”

Section: Local Monodromy On the E-partmentioning

confidence: 99%

“…We have showed in [13] that with some modifications the new object "Winger's family" parameterized all stable genus 10 curves with A 5 -symmetry. It was showed in the same paper that for a smooth member C t (t ∈ B • a point in the smooth locus of W → B) of the Winger pencil, its space of holomorphic forms H 0 (C, ω C ) is isomorphic to V ⊕ I ⊕ I = V ⊕ E as a CI-module where V is the permutation representation of dimension four and I and I are three dimensional irreducible representations.…”

Section: Introductionmentioning

confidence: 99%

On the Monodromy and Period Map of the Winger Pencil

Zi¹

2023

Preprint

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The sextic plane curves that are invariant under the standard action of the icosahedral group on the projective plane make up a pencil of genus ten curves (spanned by a sum of six lines and a three times a conic). This pencil was first considered in a note by R. M. Winger in 1925 and is nowadays named after him. We gave this a modern treatment and proved among other things that it contains essentially every smooth genus ten curve with icosahedral symmetry. We here consider the monodromy group and the period map naturally defined by the icosahedral symmetry. We showed that this monodromy group is a subgroup of finite index in SL 2 (Z[ √ 5]) and the period map brings the Winger pencil to a curve on the Hilbert modular surface SL 2 (Z[ √ 5])/H 2 .

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Bring’s curve: old and new

Braden,

Disney-Hogg

2023

European Journal of Mathematics

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Bring’s curve, the unique Riemann surface of genus 4 with automorphism group $$S_5$$ S 5 , has many exceptional properties. We review, give new proofs of, and extend a number of these including giving the complete realisation of the automorphism group for a plane curve model, identifying a new elliptic quotient of the curve and the modular curve $$X_0(50)$$ X 0 ( 50 ) , providing a complete description of the orbit decomposition of the theta characteristics, and identifying the unique invariant characteristic with the divisor of the Szegő kernel. In achieving this we have used modern computational tools in Sagemath, Macaulay2, and Maple, for which notebooks demonstrating calculations are provided.

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Monodromy and period map of the Winger pencil

Looijenga

¹

,

Zi

²

2023

Journal of London Math Soc

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The sextic plane curves that are invariant under the standard action of the icosahedral group on the projective plane make up a pencil of genus ten curves (spanned by a sum of six lines and three times a conic). This pencil was first considered in a note by R. M. Winger in 1925 and is nowadays named after him. The second author recently gave this a modern treatment and proved among other things that it contains essentially every smooth genus ten curve with icosahedral symmetry. We here show that the Jacobian of such a curve contains the tensor product of an elliptic curve with a certain integral representation of the icosahedral group. We find that the elliptic curve comes with a distinguished point of order 3, which proves that the monodromy on this part of the homology is the full congruence subgroup Γ1(3)⊂SL2(Z)$\Gamma _1(3)\subset \text{SL}_2(\mathbb {Z})$ and subsequently identify the base of the pencil with the associated modular curve. We also observe that the Winger pencil “accounts” for the deformation of the Jacobian of Bring's curve as a principal abelian fourfold with an action of the icosahedral group.

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Geometry of the Winger pencil

Cited by 4 publications

References 9 publications

On the Monodromy and Period Map of the Winger Pencil

On the Monodromy and Period Map of the Winger Pencil

Bring’s curve: old and new

Monodromy and period map of the Winger pencil

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