Hutchinson-Weber involutions degenerate exactly when the Jacobian is Comessatti

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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.