Homological Projective duality (HP-duality) theory, introduced by Kuznetsov [42], is one of the most powerful frameworks in the homological study of algebraic geometry. The main result (HP-duality theorem) of the theory gives complete descriptions of bounded derived categories of coherent sheaves of (dual) linear sections of HP-dual varieties. We show the theorem also holds for more general intersections beyond linear sections. More explicitly, for a given HP-dual pair (X, Y ), then analogue of HP-duality theorem holds for their intersections with another HP-dual pair (S, T ), provided that they intersect properly. We also prove a relative version of our main result. Taking (S, T ) to be dual linear subspaces (resp. subbundles), our method provides a more direct proof of the original (relative) HPduality theorem.

This paper studies the derived category of the Quot scheme of rank d locally free quotients of a sheaf G of homological dimension ≤ 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), Cayley's 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) Quot 2 schemes, (ii) flips from partial desingularizations of rank ≤ 2 degeneracy loci, and (iii) blowups along determinantal subschemes of codimension ≤ 4. Generalities on derived categoriesof schemes 3.2. Generators of triangulated categories 3.3. Semiorthogonal decompositions and mutations 3.4. Postnikov systems and convolutions 3.5. Closed monoidal structures 3.6. Linear categories 3.7. Relative Serre functors 3.8. Base change of linear categories 3.9. Relative Fourier-Mukai transforms 3.10. Compositions of relative Fourier-Mukai transforms 3.11. Relative exceptional collections 3.12. Grassmannian bundles Part II. Local geometry 4. Young diagrams and Grassmannians 4.1. Young diagrams and Schur functors 4.2. Borel-Bott-Weil theorem and Kapranov's collections 4.3. Mutations on Grassmannians 5. Local geometry and correspondences 5.1. The key lemma and Lascoux-type resolutions 5.2. First implications 5.3. The case d + = 1: projectivization 5.4. The case ℓ + = 1: standard flips 5.5. The case m = 1: Pirozhkov's theorem 5.6. The cases δ ≤ 3 5.7. The case d + = 2: Quot 2 -formula 5.8. The case ℓ + = 2: flips from resolving rank ≤ 2 degeneracy lociPart III. Global geometry 6. Global situation 6.1. Hom spaces 6.2. Tor-independent conditions and general procedures of base-change 6.3. Blowups along Koszul-regularly immersed centers 6.4. Cayley's trick 6.5. Projectivizations 6.6. Generalized Caylay's trick and Pirozhkov's theorem 6.7. Quot 2 -formula 108 7. Flips, flops and virtual flips 110 7.1. First results: Grassmannian flips and virtual flips 110 7.2. Standard flips revisited 112 7.3. Flips from partial desingularizations of rank ≤ 2 degeneracy loci 114 7.4. The cases rank G ≤ 3, and blowups of determinantal ideals of height ≤ 4 115 Appendix A. Relations in the Grothendieck rings of varieties 121 Appendix B. Characteristic-free results for projective bundles 123 References 127

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