Derived category of projectivization and generalized linear duality

In this note, we generalize the linear duality between vector subbundles (or equivalently quotient bundles) of dual vector bundles to coherent quotients $V \twoheadrightarrow \mathscr{L}$ considered in arXiv:1811.12525, in the framework of Kuznetsov's homological projective duality (HPD). As an application, we obtain a generalized version of the fundamental theorem of HPD for the $\mathbb{P}(\mathscr{L})$-sections and the respective dual sections of a given HPD pair.