The geometry of hereditary orders and beyond

The thesis deals with three topics linked to hereditary orders. First, we compute the class of the Brauer--Severi variety of a hereditary order in the Grothendieck ring of varieties. This leads to heuristics for a decomposition of the bounded derived category of the Brauer--Severi scheme. Second, we construct a semiorthogonal decomposition of the bounded derived category of the hereditary order using categorical absorption introduced by Kuznetsov--Shinder (2023). The last chapter is joint work with Pieter Belmans and Okke van Garderen. We study the restriction of tame orders on surfaces along central curves appealing to a dictionary between orders and stacks, and determine when such a restriction is a hereditary order on a curve. This leads to applications for noncommutative plane curves.