Derived categories of Quot schemes of locally free quotients, I

This paper studies the derived category of the Quot scheme of rank $d$ locally free quotients of a sheaf $mathscr{G}$ of homological dimension $le 1$ over a scheme $X$. In particular, we propose a conjecture about the structure of its derived category and verify the conjecture in various cases. This framework allows us to relax certain regularity conditions on various known formulae -- such as the ones for blowups (along Koszul-regular centers), Cayleys trick, standard flips, projectivizations, and Grassmannain-flips -- and supplement these formulae with the results on mutations and relative Serre functors. This framework also leads us to many new phenomena such as virtual flips, and structural results for the derived categories of (i) $mathrm{Quot}_2$ schemes, (ii) flips from partial desingularizations of $mathrm{rank}le 2$ degeneracy loci, and (iii) blowups along determinantal subschemes of codimension $le 4$.