Monodromy and period map of the Winger Pencil

TL;DR

This paper studies the monodromy and period map of the Winger pencil, a family of genus ten curves with icosahedral symmetry, revealing deep connections with elliptic curves, modular curves, and Jacobian deformations.

Contribution

It provides a detailed analysis of the monodromy, Jacobian structure, and modular interpretation of the Winger pencil, linking geometric, algebraic, and modular aspects of these symmetric curves.

Findings

The Jacobian contains a tensor product with an elliptic curve with a 3-torsion point.

Monodromy on this part is the full congruence subgroup Γ₁(3).

The base of the pencil is identified with a modular curve.

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 $\Gamma_1(3)\subset \SL_2(\Zds)$ 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.