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 ${\cal{D}}$ 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 ${\cal{D}}$ modulo the rational first integrals of ${\cal{D}}$. We prove that this group is always a finite $\mathbb{Z}/p$-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 $A^p$ and $A$, we show that the kernel of the pull-back morphism $Pic(B)\rightarrow Pic(A)$ is a finite $\mathbb{Z}/p$-vector space. In particular, if $A$ is a UFD, then the Picard group of $B$ is finite.