Companion varieties for root systems and Fermat arrangements

Abstract

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 connections to various other areas of mathematics including vector bundles , arrangements of hyperplanes , geometry of projective varieties. Our research is motivated by what is now known as the BMSS duality, which is a new way of deriving projective varieties out of already constructed ones. 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. In the case of configurations of points associated to root systems, the geometry of their companions is also described.