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INVARIANT HYPERSURFACES
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 21 / Issue 2 / March 2022
- Published online by Cambridge University Press:
- 17 August 2020, pp. 713-739
- Print publication:
- March 2022
-
- Article
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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 
$\unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{2}:Z\rightarrow X$ are dominant rational maps from an (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 
$\unicode[STIX]{x1D719}_{1}$ and 
$\unicode[STIX]{x1D719}_{2}$ agree. Then there is a nonconstant rational function 
$g$ on 
$X$ such that 
$g\unicode[STIX]{x1D719}_{1}=g\unicode[STIX]{x1D719}_{2}$. In the case where 
$Z$ 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.
