Fullness of the Kuznetsov–Polishchuk Exceptional Collection for the Spinor Tenfold

Abstract

Kuznetsov and Polishchuk provided a general algorithm to construct exceptional collections of maximal length for homogeneous varieties of type A, B, C, D. We consider the case of the spinor tenfold and we prove that the corresponding collection is full, i.e. it generates the whole derived category of coherent sheaves. We also verify strongness and purity of such collection. As a step of the proof, we construct some resolutions of homogeneous vector bundles which might be of independent interest.