Invariant hypersurfaces

Abstract

The following theorem, which includes as very special cases results of Jouanolou and Hrushovski on algebraic $D$-varieties on the one hand, and of Cantat on rational dynamics on the other, is established: Working over a field of characteristic zero, suppose $\ddot{I}{•}_1,\ddot{I}{•}_2:Z\to X$ are dominant rational maps from a (possibly nonreduced) irreducible scheme $Z$ of finite-type to an algebraic variety $X$, with the property that there are infinitely many hypersurfaces on $X$ whose scheme-theoretic inverse images under $\ddot{I}{•}_1$ and $\ddot{I}{•}_2$ agree. Then there is a nonconstant rational function $g$ on $X$ such that $g\ddot{I}{•}_1=g\ddot{I}{•}_2$. In the case when $Z$ is also reduced the scheme-theoretic inverse image can be replaced by the proper transform. A partial result is obtained in positive characteristic. Applications include an extension of the Jouanolou-Hrushovski theorem to generalised algebraic $\mathcal{D}$-varieties and of Cantat's theorem to self-correspondences.

Final version following minor changes suggested by the referee