Derived categories of projectivizations and flops

Abstract

We prove a generalization of Orlov's projectivization formula for the derived category Dcohb(P(E)), where E does not need to be a vector bundle; Instead, E is a coherent sheaf which locally admits two-step resolutions. As a special case, this also gives Orlov's generalized universal hyperplane section formula. As applications, (i) we obtain a blowup formula for blowup along codimension two Cohen-Macaulay subschemes, (ii) we obtain new “flop-flop=twist” results for a large class of flops obtained by crepant resolutions of degeneracy loci. As another consequence, this gives a perverse schober on C. (iii) we give applications of above results to symmetric powers of curves and Θ-flops, following Toda [79].