COMPANION VARIETIES FOR HESSE, HESSE UNION DUAL HESSE ARRANGEMENTS

. Unexpected hypersurface is a name given an element to some particular linear system introduced around in [2], motivated by work of Di Gennaro, Ilardi and Vall`es and of Faenzi and Vall`es, and it is a field of great study since then. It attracts many people because of their close tights to various other areas of mathematics including vector bundles, arrangements of hyperplanes, geometry of projective varieties, etc. In this paper we continue the study about BMSS duality, of [3]. In [6], the authors introduced the concept of unexpected hypersurfaces and they explain 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 revisit the configuration of points associated to Hesse arrangement, Hesse union dual Hesse arrangement and we study the geometry of the associated varieties and their companions.