Derived categories of fibrations of quintic del Pezzo surfaces

I will provide two approaches for fining a semiorthogonal decomposition of the derived category of fibrations of quintic del Pezzo surfaces with RDP singularities. Similar to Kuznetsov’s work on sextic del Pezzo surfaces, the components of the semiorthogonal decomposition can be interpreted as the moduli spaces of semistable sheaves on fibers with fixed Hilbert polynomials. Alternatively, there is a rank 2 vector bundle on a quintic del Pezzo surface with RDP singularities that embeds the surface as a linear section of a Grassmannian. The semiorthogonal decomposition can be obtained by applying Homological Projective Duality. This is work in progress.