  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.
