Exceptional collections of sheaves on quadrics in positive characteristic

Abstract

On quadrics in large positive characteristic we construct an exceptional collection of sheaves from the G1P-Verma module associated to the Frobenius direct image of the structure sheaf of the quadric. They are all locally free of finite rank and defined over Z, producing Kapranov’s collection over C by base change. We also determine on a general homogeneous projective variety in large positive characteristic a direct summand of the Frobenius direct image of the structure sheaf, which is “dual” to the celebrated Frobenius splitting of Mehta and Ramanathan.