{
  "id": "1802.01165",
  "version": "v1",
  "published": "2018-02-04T17:45:15.000Z",
  "updated": "2018-02-04T17:45:15.000Z",
  "title": "Ultrametric properties for valuation spaces of normal surface singularities",
  "authors": [
    "Evelia García Barroso",
    "Pedro González Pérez",
    "Patrick Popescu-Pampu",
    "Matteo Ruggiero"
  ],
  "comment": "48 pages, 15 figures",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $L$ be a fixed branch -- that is, an irreducible germ of curve -- on a normal surface singularity $X$. If $A,B$ are two other branches, define $u_L(A,B) := \\dfrac{(L \\cdot A) \\: (L \\cdot B)}{A \\cdot B}$, where $A \\cdot B$ denotes the intersection number of $A$ and $B$. Call $X$ arborescent if all the dual graphs of its resolutions are trees. In a previous paper, the first three authors extended a 1985 theorem of P{\\l}oski by proving that whenever $X$ is arborescent, the function $u_L$ is an ultrametric on the set of branches on $X$ different from $L$. In the present paper we prove that, conversely, if $u_L$ is an ultrametric, then $X$ is arborescent. We also show that for any normal surface singularity, one may find arbitrarily large sets of branches on $X$, characterized uniquely in terms of the topology of the resolutions of their sum, in restriction to which $u_L$ is still an ultrametric. Moreover, we describe the associated tree in terms of the dual graphs of such resolutions. Then we extend our setting by allowing $L$ to be an arbitrary semivaluation on $X$ and by defining $u_L$ on a suitable space of semivaluations. We prove that any such function is again an ultrametric if and only if $X$ is arborescent, and without any restriction on $X$ we exhibit special subspaces of the space of semivaluations in restriction to which $u_L$ is still an ultrametric.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-04T17:45:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14J17",
      "14B05",
      "32S05"
    ],
    "keywords": [
      "normal surface singularity",
      "ultrametric properties",
      "valuation spaces",
      "dual graphs",
      "arborescent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}