A tale of three GIT problems

A general net of quadric surfaces, together with a choice of a base point, defines a net of plane cubics via Gale duality. To both nets, one can also naturally associate the same smooth plane quartic. In this talk, I will report on joint work with M. Hattori and T. Papazachariou, concerning a generalisation of this classical threefold cycle of correspondences. I will explain how, by extending these correspondences, one can obtain a complete criterion for GIT stability of the
three underlying geometric objects using a birational-geometric method.