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arXiv · 2109.01810

Monodromy and period map of the Winger Pencil

Eduard Looijenga·Yunpeng Zi
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Abstract

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. 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 $3$, prove that the monodromy on this part of the homology is the full congruence subgroup $Γ_1(3)\subset \SL_2(\Zds)$ 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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Eduard Looijenga, Yunpeng Zi. 2022-12-10. Monodromy and period map of the Winger Pencil. https://arxiv.org/abs/2109.01810

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