Categorical Plücker Formula and Homological Projective Duality

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) HP-duality theorem.