Student AG Seminar: Understanding the derived category of linear GIT quotients by rank 2 groups

In this talk, which is mainly based on the paper https://arxiv.org/abs/2202.12876 by 
Daniel Halpern-Leistner, Kimoi Kemboi, I will try to investigate the derived category of linear GIT quotients by rank 2 groups, such as \$\operatorname{GL}_2\$. The main focus of the talk is to understand what is called the "window category" which will give us an exceptional collection. Under very special conditions this will also be a full exceptional collection.