### Abstract

We show that if ϕ: X → X is an automorphism of a smooth projective variety and D ⊂ X is an irreducible divisor for which the set [d ∈ D: ϕ^{n}(d) ∈2 D for some nonzero n] is not Zariski dense in D, then (X; ϕ) admits an equivariant rational fibration to a curve. As a consequence, we show that certain blow-ups (for example, blow-ups in high codimension) do not alter the finiteness of 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 nonreduced analog of the dynamical Mordell-Lang conjecture. Namely, let ϕ: X → 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) ⊆ Z] is the union of a finite set and finitely many residue classes; its modulus is bounded in terms of the geometry of Y .

Original language | English (US) |
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Pages (from-to) | 1-25 |

Number of pages | 25 |

Journal | Algebraic Geometry |

Volume | 6 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2019 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Geometry and Topology

### Cite this

*Algebraic Geometry*,

*6*(1), 1-25. https://doi.org/10.14231/AG-2019-001