Hutchinson–Weber Involutions Degenerate Exactly when the Jacobian is Comessatti

Abstract

We consider Jacobian Kummer surface $X$ of a genus two curve $C$. We prove that Hutchinson-Weber involution on $X$ degenerates if and only if Jacobian $J(C)$ is Comessatti. Also we give several conditions equivalent to this, which include classical theorem of Humbert. The key notion is Weber hexad. We include explanation of them and discuss dependence between 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 computation of patching subgroup for generic Hutchinson-Weber involutions.