Derived category of projectivization and flops

摘要: 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 coherent sheaf on regular scheme which locally admits two-step resolutions. Second, that "flop-flop=twist" results hold flops obtained by two different Springer-type resolutions of determinantal hypersurface. This also gives sequence higher dimensional examples present perverse schobers, and provide further evidences proposal Bondal-Kapranov-Schechtman [BKS,KS]. Applications to symmetric powers curves, Abel-Jacobi maps $\Theta$-flops following Toda are discussed.