Derived categories of quintic del Pezzo fibrations

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