{
  "id": "1703.03294",
  "version": "v1",
  "published": "2017-03-09T15:30:50.000Z",
  "updated": "2017-03-09T15:30:50.000Z",
  "title": "Veronese varieties contained in hypersurfaces",
  "authors": [
    "Jason Michael Starr"
  ],
  "comment": "19 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "Alex Waldron proved that for sufficiently general degree $d$ hypersurfaces in projective $n$-space, the Fano scheme parameterizing $r$-dimensional linear spaces contained in the hypersurface is nonempty precisely for the degree range $n\\geq N_1(r,d)$ where the \"expected dimension\" $f_1(n,r,d)$ is nonnegative, in which case $f_1(n,r,d)$ equals the (pure) dimension. Using work by Gleb Nenashev, we prove that for sufficiently general degree $d$ hypersurfaces in projective $n$-space, the parameter space of $r$-dimensional $e$-uple Veronese varieties contained in the hypersurface is nonempty of pure dimension equal to the \"expected dimension\" $f_e(n,r,d)$ in a degree range $n\\geq \\widetilde{N}_e(r,d)$ that is asymptotically sharp. Moreover, we show that for $n\\geq 1+N_1(r,d)$, the Fano scheme parameterizing $r$-dimensional linear spaces is irreducible.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-09T15:30:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14C05",
      "14J70"
    ],
    "keywords": [
      "hypersurface",
      "dimensional linear spaces",
      "sufficiently general degree",
      "fano scheme parameterizing",
      "degree range"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}