{
  "id": "0806.1364",
  "version": "v2",
  "published": "2008-06-09T02:01:48.000Z",
  "updated": "2008-11-20T16:54:31.000Z",
  "title": "Isotriviality is equivalent to potential good reduction for endomorphisms of ${\\mathbb P}^N$ over function fields",
  "authors": [
    "Clayton Petsche",
    "Lucien Szpiro",
    "Michael Tepper"
  ],
  "comment": "Changes in this version: moved some preliminary material on non-archimedean fields to section 2; clarified the geometric proof of Theorem 1; replaced our proof of Prop. 2(c)--which had a gap in it--with a reference to the proof by Fakhruddin; corrected several small errors and typos, and added some new references",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "Let $K=k(C)$ be the function field of a complete nonsingular curve $C$ over an arbitrary field $k$. The main result of this paper states that a morphism $\\phi:{\\mathbb P}^N_K\\to{\\mathbb P}^N_K$ is isotrivial if and only if it has potential good reduction at all places $v$ of $K$; this generalizes results of Benedetto for polynomial maps on ${\\mathbb P}^1_K$ and Baker for arbitrary rational maps on ${\\mathbb P}^1_K$. We offer two proofs: the first uses algebraic geometry and geometric invariant theory, and it is new even in the case N=1. The second proof uses non-archimedean analysis and dynamics, and it more directly generalizes the proofs of Benedetto and Baker. We will also give two applications. The first states that an endomorphism of ${\\mathbb P}^N_K$ of degree at least two is isotrivial if and only if it has an isotrivial iterate. The second gives a dynamical criterion for whether (after base change) a locally free coherent sheaf ${\\mathcal E}$ of rank $N+1$ on $C$ decomposes as a direct sum ${\\mathcal L}\\oplus...\\oplus{\\mathcal L}$ of $N+1$ copies of the same invertible sheaf ${\\mathcal L}$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-11-20T16:54:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14G99",
      "14H05"
    ],
    "keywords": [
      "function field",
      "endomorphism",
      "isotriviality",
      "equivalent",
      "locally free coherent sheaf"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0806.1364P"
    }
  }
}