On the Derived Categories of Degree d Hypersurface Fibrations

We provide descriptions of the derived categories of degree $d$ hypersurface fibrations which generalize a result of Kuznetsov for quadric fibrations and give a relative version of a well-known theorem of Orlov. Using a local generator and Morita theory, we re-interpret the resulting matrix factorization category as a derived-equivalent sheaf of dg-algebras on the base. Then, applying homological perturbation methods, we obtain a sheaf of $A_\infty$-algebras which gives a new description of homological projective duals for (relative) $d$-Veronese embeddings, recovering the sheaf of Clifford algebras obtained by Kuznetsov in the case when $d=2$.