On the Derived Categories of Degree d Hypersurface Fibrations

TL;DR

This paper extends the understanding of derived categories for degree d hypersurface fibrations, generalizing known results for quadrics and providing new descriptions of homological projective duals using advanced algebraic techniques.

Contribution

It introduces a new framework for describing derived categories of hypersurface fibrations of arbitrary degree d, generalizing Kuznetsov's quadric case and relating to homological projective duality.

Findings

Derived categories described via sheaves of dg-algebras and A_infinity-algebras.

Reinterpretation of matrix factorizations in the context of hypersurface fibrations.

Recovery of Kuznetsov's Clifford algebra sheaf in the quadratic case.

Abstract

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 $d=2$.