G\"opel Varieties

TL;DR

This paper explores a family of hypersurfaces related to G"opel varieties, connecting their geometric properties with algebraic structures via Jordan-Vinberg theory and complex reflection groups.

Contribution

It introduces a new framework for understanding G"opel varieties through parametrization by G"opel type varieties in homogeneous spaces using complex reflection groups.

Findings

Identification of G"opel varieties within a broader family of hypersurfaces.

Parametrization of these hypersurfaces using Jordan-Vinberg theory.

Connection established between geometric properties and algebraic group actions.

Abstract

We show that the Coble hypersurfaces, uniquely characterized by the remarkable property that their singular loci are an abelian surface and a Kummer threefold, respectively, belong to a family of hypersurfaces exhibiting similar behavior, but defined in various types of homogeneous spaces. With the help of Jordan-Vinberg theory, we show how these hypersurfaces can be parametrized by G{\"o}pel type varieties inside projectivized representations of complex reflection groups.