[BIMSA Topology Seminar] Monodromy and period map of the Winger Pencil

Abtract：

We investigate the moduli space of genus 10 curves that are endowed with a faithful action of the icosahedral group $\Acal_5$. We will show among other things that this moduli space essentially consists two copies of the pencil of plane sextics introduced by Winger, an American mathematician, in 1924 with the unique unstable member (a triple conic) replaced by a smooth non-planar curve.
We also prove that the Jacobian of a smooth member of the Winger pencil 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.
Except those we also observed 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.