Companion varieties for root systems and Fermat arrangements

TL;DR

This paper explores the construction and geometric properties of companion varieties related to root systems and Fermat arrangements, extending the concept of companion surfaces and examining their connections to unexpected hypersurfaces.

Contribution

It extends the concept of companion surfaces to various root systems and Fermat arrangements, analyzing their associated varieties and geometric relations.

Findings

Revisited point configurations for root systems and Fermat arrangements.

Studied the geometry of associated varieties and their companions.

Extended the construction of companion surfaces beyond previous cases.

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 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.