  Unexpected hypersurfaces are a brand name for some special linear systems.
They were introduced around 2017 and are a field of intensive study since then.
They attracted a lot of attention because of their close tights to various
other areas of mathematics including vector bundles, arrangements of
hyperplanes, geometry of projective varieties. Our research is motivated by the
what is now known as the BMSS duality, which is a new way of deriving
projective varieties out of already constructed. The last author coined the
concept of companion surfaces in the setting of unexpected curves admitted by
the $B_3$ root system. Here we extend this construction in various directions.
We revisit the configurations of points associated to either root systems or to
Fermat arrangements and we study the geometry of the associated varieties and
their companions.
