# Invariant hypersurfaces

## TL;DR

This paper proves a general theorem about invariant hypersurfaces under rational maps, unifying and extending results in algebraic dynamics and differential algebra, with applications to algebraic D-varieties and rational dynamics.

## Contribution

It establishes a broad invariant hypersurface theorem that generalizes previous results by Jouanolou, Hrushovski, and Cantat, applicable over characteristic zero fields and partially in positive characteristic.

## Key findings

- Existence of a nonconstant rational function relating two maps with invariant hypersurfaces.
- Extension of Jouanolou-Hrushovski theorem to generalized algebraic D-varieties.
- Extension of Cantat's theorem to self-correspondences.

## 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 $\phi_1,\phi_2: Z \to X$ 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 $\phi_1$ and $\phi_2$ agree. Then there is a nonconstant rational function $g$ on $X$ such that $g\phi_1=g\phi_2$. 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.