Let A be an integral k-algebra of finite type over an algebraically
closed field k of characteristic p>0. Given a collection D of
k-derivations on A, that we interpret as algebraic vector fields on
X=Spec(A), we study the group spanned by the hypersurfaces V(f) of X
invariant for D modulo the rational first integrals of D.
We prove that this group is always a finite Z/p-vector space, and we
give an estimate for its dimension. This is to be related to the results of
Jouanolou and others on the number of hypersurfaces invariant for a foliation
of codimension 1. As an application, given a k-algebra B between Ap and
A, we show that the kernel of the pull-back morphism Pic(B)→Pic(A) is a finite Z/p-vector space. In particular, if A is a
UFD, then the Picard group of B is finite.Comment: 16 page