Hutchinson-Weber involutions degenerate exactly when the Jacobian is Comessatti (Q409063)

An abelian surface is called \textit{Comessatti} if \(A\) has a real multiplication in \(\mathcal{O}_{\mathbb{Q}(\sqrt{5})}\). \textit{HW (Hutchinson-Weber)-involution} is an involution acting on \(X_W\) and we say HW-involution degenerates if it acquires fixed loci.NEWLINENEWLINEIt is proved (algebraically in this paper) that for any Jacobian Kummer surface \(X = \mathrm{Kum}(J(\Sigma))\) associated to a genus-2 curve \(\Sigma\), there exists a Weber hexad \(W\) (certain choice of six points) such that \(X\) is birational to \(X_W\). The first main theorem of the paper under review is the study of the relations among the following three notions:NEWLINENEWLINE 1) A Hessian quartic surface \(H_C\) of a smooth projective cubic surface \(C\) that admits exactly 11 singular points;NEWLINENEWLINE 2) Comessatti Jacobican \(J(\Sigma)\) of genus-2 curve \(\Sigma\);NEWLINENEWLINE 3) Weber hexads; and also a relation with the HW involution. Finally the moduli spaces of Kummer surfaces which areNEWLINENEWLINE 1) Jacobian;NEWLINENEWLINE 2) Jacobian with Weber hexad;NEWLINENEWLINE 3) Comessatti Jacobian are described explicitly in terms of elements in the transcendental lattice of the Kummer surface.NEWLINENEWLINEThe author remarks that the case of abelian surfaces being product-type of two elliptic curves is a further research to study.