Dynamical Mordell-Lang and Automorphisms of Blow-ups

Abstract

We show that if $ϕ: X \to X$ is an automorphism of a smooth projective variety and $D \subset X$ is an irreducible divisor for which the set of $d$ in $D$ with $ϕ^n(d)$ in $D$ for some nonzero $n$ is not Zariski dense, then $(X, ϕ)$ admits an equivariant rational fibration to a curve. As a consequence, we show that certain blowups (e.g. blowups in high codimension) do not alter the finiteness of $\textrm{Aut}(X)$, extending results of Bayraktar-Cantat. We also generalize results of Arnol'd on the growth of multiplicities of the intersection of a variety with the iterates of some other variety under an automorphism. These results follow from a non-reduced analogue of the dynamical Mordell-Lang conjecture. Namely, let $ϕ: X \to X$ be an étale endomorphism of a smooth projective variety $X$ over a field $k$ of characteristic zero. We show that if $Y$ and $Z$ are two closed subschemes of $X$, then the set $A_ϕ(Y,Z) = \{n : ϕ^n(Y) \subseteq Z\}$ is the union of a finite set and finitely many residue classes, whose modulus is bounded in terms of the geometry of $Y$.