Hutchinson-Weber involutions degenerate exactly when the Jacobian is Comessatti

TL;DR

This paper characterizes when Hutchinson-Weber involutions on Jacobian Kummer surfaces degenerate, linking it to the Jacobian being Comessatti, and explores related moduli spaces and classical conditions.

Contribution

It establishes a precise criterion for degeneration of Hutchinson-Weber involutions in terms of Comessatti Jacobians and relates this to classical theorems and Weber hexads.

Findings

Degeneration occurs iff the Jacobian is Comessatti.

Provides conditions equivalent to degeneration including Humbert's theorem.

Describes the moduli space structure and computes the patching subgroup.

Abstract

We consider the Jacobian Kummer surface $X$ of a genus two curve $C$. We prove that the Hutchinson-Weber involution on $X$ degenerates if and only if the Jacobian $J(C)$ is Comessatti. Also we give several conditions equivalent to this, which include the classical theorem of Humbert. The key notion is the Weber hexad. We include explanation of them and discuss the dependence between the conditions of main theorem for various Weber hexads. It results in "the equivalence as dual six". We also give a detailed description of relevant moduli spaces. As an application, we give a conceptual proof of the computation of the patching subgroup for generic Hutchinson-Weber involutions.