Invariant hypersurfaces for derivations in positive characteristic

Let AA be an integral kk-algebra of finite type over an algebraically closed field kk of characteristic p>0p>0. Given a collection DD of kk-derivations on AA, that we interpret as algebraic vector fields on X=Spec(A), we study the group spanned by the hypersurfaces V(f)V(f) of XX invariant under DD modulo the rational first integrals of DD. We prove that this group is always a finite dimensional FpFp-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 under a foliation of codimension 1. As a application, given a kk-algebra BB between ApAp and AA, we show that the kernel of the pull-back morphism Pic(B)→Pic(A) is a finite FpFp-vector space. In particular, if AA is a UFD, then the Picard group of BB is finite.