Derived category of projectivization and flops

In this paper, we first show a projectivization formula for the derived category $D^b_{rm coh} (mathbb{P}(mathcal{E}))$, where $mathcal{E}$ is a coherent sheaf on a regular scheme which locally admits two-step resolutions. Second, we show that flop-flop=twist results hold for flops obtained by two different Springer-type resolutions of a determinantal hypersurface. This also gives a sequence of higher dimensional examples of flops which present perverse schobers, and provide further evidences for the proposal of Bondal-Kapranov-Schechtman [BKS,KS]. Applications to symmetric powers of curves, Abel-Jacobi maps and $Theta$-flops following Toda are also discussed.