An invariant for hypersurfaces in prime characteristic

A hypersurface of order (n + 1)(p ^h - 1) in projective space of dimension n of prime characteristic p has an invariant monomial. This implies that a hypersurface of order (n+1)(p ^h-1)-1 determines an invariant point. A hypersurface of order d < n+ 1 in a projective space of dimension n of characteristic two has an invariant set of subspaces of dimension d-1 determined by one linear condition on the Grassmann coordinates of the dual subspaces.