{
  "id": "1407.6945",
  "version": "v2",
  "published": "2014-07-25T15:45:49.000Z",
  "updated": "2014-08-20T15:24:30.000Z",
  "title": "Low degree hypersurfaces of projective toric varieties defined over a $C_1$ field have a rational point",
  "authors": [
    "Robin Guilbot"
  ],
  "comment": "46 pages. This is the long version of the article, with quite a lot of preliminaries aimed at non (toric) geometers. Changes from v1 : Statement and proof of the core theorem simplified, section about decomposition of toric rational contractions removed + minor corrections",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "Quasi algebraically closed fields, or $C_1$ fields, are defined in terms of a low degree condition. Namely, the field $K$ is $C_1$ if every degree $d$ hypersurface of the projective space $\\mathbb{P}_K^n$ contains a $K$-point as soon as $d\\leq n$. In this article we define a notion of low toric degree generalizing this condition for hypersurfaces of simplicial projective split toric varieties. This allows us to prove a particular case of the $C_1$ conjecture of Koll\\'{a}r, Lang and Manin : any smooth separably rationally connected variety that can be embedded as such a hypersurface over a $C_1$ field has a rational point. Our results are based on the fact that the ambient toric varieties are Mori Dream Spaces : they are naturally endowed with homogeneous coordinates and their Minimal Model Program works in all cases.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-25T15:45:49.000Z",
      "abstract": "Quasi algebraically closed fields, or $C_1$ fields, are defined in terms of a low degree condition. Namely, the field $K$ is $C_1$ if every degree $d$ hypersurface of the projective space $\\mathbb{P}_K^n$ contains a $K$-point as soon as $d\\leq n$. In this article we define a notion of low toric degree generalizing this condition for hypersurfaces of simplicial projective split toric varieties. This allows us to prove a particular case of the $C_1$ conjecture of Koll\\'ar, Lang and Manin : any smooth separably rationally connected variety that can be embedded as such a hypersurface over a $C_1$ field has a rational point. Our results are based on the fact that the ambient toric varieties are Mori Dream Spaces : they are naturally endowed with homogeneous coordinates and their Minimal Model Program works in all cases.",
      "comment": "50 pages. This is the long version of the article, with quite a lot of preliminaries aimed at non (toric) geometers",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-08-20T15:24:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M25",
      "11G99",
      "14G05",
      "14E30"
    ],
    "keywords": [
      "projective toric varieties",
      "low degree hypersurfaces",
      "rational point",
      "projective split toric varieties",
      "separably rationally connected variety"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.6945G"
    }
  }
}