Invariant hypersurfaces for derivations in positive characteristic

Abstract

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 under D modulo the rational first integrals of D . We prove that this group is always a finite dimensional F 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 under a foliation of codimension 1. As a application, given a k -algebra B between A p and A , we show that the kernel of the pull-back morphism Pic ( B ) → Pic ( A ) is a finite F p -vector space. In particular, if A is a UFD, then the Picard group of B is finite.