Geometry of the Wiman Pencil, I: Algebro-Geometric Aspects

TL;DR

This paper revisits the Wiman pencil of genus 6 curves, exploring its geometric properties with modern methods to deepen understanding of its algebraic and geometric structure.

Contribution

It provides a modern, conceptual analysis of the Wiman pencil, translating explicit equations into geometric objects and offering new insights into its structure.

Findings

Characterization of the Wiman pencil in geometric terms

Identification of symmetries and special properties of the curves

Connection between explicit equations and geometric objects

Abstract

In 1981 W.L. Edge discovered and studied a pencil $\mathcal{C}$ of highly symmetric genus $6$ projective curves with remarkable properties. Edge's work was based on an 1895 paper of A. Wiman. Both papers were written in the satisfying style of 19th century algebraic geometry. In this paper and its sequel [FL], we consider $\mathcal{C}$ from a more modern, conceptual perspective, whereby explicit equations are reincarnated as geometric objects.