The coherent derived category of an algebraic variety is an important invariants of the variety, and is supposed to be closely  related to its birational geometry. We start by reviewing basic definitions and results on derived categories in this direction, including DK conjecture, and examples of standard flops and flips.

Then I will talk about a recent joint work with Conan Leung on the structure of derived categories of projectvization of a coherent sheaf which locally admits two step resolutions. In the case when the rank of coherent sheaf is zero, we show that ``flop—flop=twist” results hold for flops obtained by different resolutions of determinantal hypersurface.

This provides higher dimensional examples of flops which present Perverse schobers over a disc.