{
  "id": "1604.08216",
  "version": "v1",
  "published": "2016-04-27T20:00:19.000Z",
  "updated": "2016-04-27T20:00:19.000Z",
  "title": "Dynamical Mordell-Lang and Automorphisms of Blow-ups",
  "authors": [
    "John Lesieutre",
    "Daniel Litt"
  ],
  "comment": "26 pages, 2 figures; comments appreciated",
  "categories": [
    "math.AG",
    "math.DS",
    "math.NT"
  ],
  "abstract": "We show that if $\\phi : X \\to X$ 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 $\\phi^n(d)$ in $D$ for some nonzero $n$ is not Zariski dense, then $(X, \\phi)$ 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 $\\phi : X \\to X$ be an \\'etale 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 $A_\\phi(Y,Z) = \\{n : \\phi^n(Y) \\subseteq Z\\}$ is the union of a finite set and finitely many residue classes, whose modulus is bounded in terms of the geometry of $Y$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-27T20:00:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "automorphism",
      "smooth projective variety",
      "equivariant rational fibration",
      "high codimension",
      "zariski dense"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}