Companion varieties for Hesse, Hesse union dual Hesse arrangements

Unexpected hypersurface is a name given an element to some particular linear system introduced by Cook, Harbourne, Migliore and Nagel, motivated by work of Di Gennaro, Ilardi and Vallès and of Faenzi and Vallès, and it is a field of great study since then. It attracts many people because of their close ties to various other areas of mathematics including vector bundles, arrangements of hyperplanes, geometry of projective varieties, etc. Harbourne, Migliore, Nagel and Teitler introduced the concept of unexpected hypersurfaces and explained the so-called BMSS duality showing that unexpected curves are in some sense dual to their tangent cones at their singular point. In this paper, we continue the study of BMSS duality. We revisit the configuration of points associated to Hesse arrangement and Hesse union dual Hesse arrangement, and we study the geometry of the associated varieties and their companions.