Derived Categories of Quintic Del Pezzo Fibrations

TL;DR

This paper establishes a semiorthogonal decomposition for the derived category of quintic del Pezzo surface fibrations with rational Gorenstein singularities, revealing a structure involving the base and a degree 5 scheme.

Contribution

It introduces two novel methods for constructing semiorthogonal decompositions of derived categories for these fibrations, expanding the understanding of their categorical structure.

Findings

Decomposition includes components equivalent to the base's derived category.

A non-trivial component is equivalent to a degree 5 scheme over the base.

Two methods for constructing the decomposition are demonstrated.

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.