The Hasse-Witt-matrix of a
projective hypersurface defined over a perfect field k of characteristic p is studied
using an explicit description of the Cartier-operator. We get the following
applications. If L is a linear variety of dimension n + 1 and X a generic hypersurface
of degree d, which divides p− 1, then the Frobenius-operator ℱ on Hn(X ⋅L;𝒪L⋅Y)
is invertible.