Examples of differential geometric behaviour of projective varieties in positive characteristic

Abstract

Here we study three examples of differential geometric behaviour of projective varieties in positive characteristic: (1) the classification of smooth surfaces in P2n+1 whose m-th osculating spaces have everywhere dimension 2m (1 ≤ m ≤ n); (2) hypersurfaces with Hessian rank 0; (3) singular hypersurfaces in weighted projective spaces whose tangent sheaf is locally free and a subbundle of the restricted tangent bundle.

Keywords

• positive characteristic
• osculating spaces
• hypersurfaces
• weighted projective space
• tangent sheaf
• rational scroll
• order of contact
• ruled surface
• rational normal scroll
• hyperosculation