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INVARIANT HYPERSURFACES

Published online by Cambridge University Press:  17 August 2020

Jason Bell
Affiliation:
University of Waterloo, Department of Pure Mathematics, 200 University Avenue West, Waterloo, OntarioN2L 3G1, Canada (jpbell@uwaterloo.ca; rmoosa@uwaterloo.ca)
Rahim Moosa
Affiliation:
University of Waterloo, Department of Pure Mathematics, 200 University Avenue West, Waterloo, OntarioN2L 3G1, Canada (jpbell@uwaterloo.ca; rmoosa@uwaterloo.ca)
Adam Topaz
Affiliation:
Mathematical and Statistical Sciences, University of Alberta, 632 Central Academic Building, Edmonton, AlbertaT6G 2G1, Canada (topaz@ualberta.ca)
Corresponding

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 $\unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{2}:Z\rightarrow X$ 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.

Type
Research Article
Copyright
© The Author(s) 2020. Published by Cambridge University Press

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References

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