{
  "id": "0807.4243",
  "version": "v1",
  "published": "2008-07-26T15:51:57.000Z",
  "updated": "2008-07-26T15:51:57.000Z",
  "title": "Powers of Ideals and Fibers of Morphisms",
  "authors": [
    "David Eisenbud",
    "Joe Harris"
  ],
  "categories": [
    "math.AG",
    "math.AC"
  ],
  "abstract": "Let X\\subset PP^n be a projective scheme over a field, and let phi:X --> Y be a finite morphism. Our main result is a formula in terms of global data for the maximum of the Castelnuovo-Mumford regularity of the fibers of \\phi, considered as subschemes of \\PP^n. From an algebraic point of view, our formula is related to the theorem of Cutkosky-Herzog-Trung and Kodiyalam showing that for any homogeneous ideal I in a standard graded algebra S, the regularity of I^t can be written as dt+\\epsilon for some non-negative integers d, \\epsilon, and all large t. In the special case where I contains a power of S_+ and is generated by forms of a single degree, our formula gives an interpretation of \\epsilon: it is one less than the maximum regularity of a fiber of the morphism associated to I. These formulas have strong consequences for ideals generated by generic forms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-07-26T15:51:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14N05",
      "14B05",
      "14J40",
      "13D02"
    ],
    "keywords": [
      "algebraic point",
      "castelnuovo-mumford regularity",
      "main result",
      "finite morphism",
      "generic forms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0807.4243E"
    }
  }
}