PCF all the way down
Let k be a field of characteristic zero and let f: X → X be a finite surjective map of integral k-schemes of finite type. By the critical locus of f we mean the image in X of the critical locus of the map induced on the normalization of X. By the branch locus we mean the image of the critical locus under f. We say that f is post-critically finite (PCF) if, for every irreducible divisor D contained in the branch locus, the forward orbit f(D) ∪ f^2(D) ∪ … is algebraic, that is, a finite union of subvarieties, taken as a closed subscheme with its reduced induced structure. We prove that if f is…
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