Submission date: 20 December 2018

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 f_1, f_2: Z → 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 f_1 and f_2 agree. Then there is a nonconstant rational function g on X such that gf_1=gf_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 D-varieties and of Cantat's theorem to self-correspondences.

Mathematics Subject Classification: Primary 14E99, Secondary 12H05 and 12H10

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Full text arXiv 1812.08346: pdf, ps.