2006/02/15 by Philippe Bonnet, Bonnet, Philippe
Mathematics · #13N15 #14J70 #14J99 #14R99 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AC #math.AG #msc:13N15 #msc:14J70 #msc:14J99 #msc:14R99
paper · pdf · doi:10.48550/arxiv.math/0602338
16 pages
arxiv created 2006/02/15 · openalex publication_date 2006/02/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let A be an integral k-algebra of finite type over an algebraically closed field k of characteristic p>0. Given a collection \calD 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 \calD modulo the rational first integrals of \calD. 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.