Derived categories of quintic del Pezzo fibrations

Abstract

We provide a semiorthogonal decomposition for the derived category of fibrations of quintic del Pezzo surfaces with rational Gorenstein singularities. There are three components, two of which are equivalent to the derived categories of the base and the remaining non-trivial component is equivalent to the derived category of a flat and finite of degree 5 scheme over the base. We introduce two methods for the construction of the decomposition. One is the moduli space approach following the work of Kuznetsov on the sextic del Pezzo fibrations and the components are given by the derived categories of fine relative moduli spaces. The other approach is that one can realize the fibration as a linear section of a Grassmannian bundle and apply homological projective duality.

Nodal quintic del Pezzo threefolds and their derived categories

“…In dimension 2, Karmazyn-Kuznetsov-Sinder [5] prove that a projective Gorenstein toric surface has a SOD of this kind if and only if it has the trivial Brauer group. Moreover, their method also provides some non-toric examples: the du Val sextic del Pezzo surfaces [9] and the du Val quintic del Pezzo surfaces [15]. In dimension 3, Kawamata [7] provides two such examples, the nodal quadric threefold and the nodal sextic del Pezzo threefold, by investigating the derived category of a threefold with an ordinary double point.…”

Fano fourfolds of K3 type

“…dP 5 fibrations. If → is a dP 5 fibration, the Kuznetsov component is equivalent to D b ( ) for → a flat degree 5 cover [Xie21]. In our cases, is either ℙ 2 or a quadric, and the question of being (related to) a K3 surface remains open.…”

Factorization centers in dimension two and the Grothendieck ring of varieties

“…The Galois sets forming A X naturally appear in the Chow motive of X [10, (9) and the following Remark] and as semiorthogonal components in the derived category of the corresponding surface [4], [2, Propositions 9.8, 10.1]. Furthermore, singular versions of those k-algebras show up in the study of the derived categories of the corresponding singular del Pezzo surfaces [21], [19], [37].…”