{
  "id": "1603.03294",
  "version": "v1",
  "published": "2016-03-10T15:09:03.000Z",
  "updated": "2016-03-10T15:09:03.000Z",
  "title": "On homomorphisms between Cremona groups",
  "authors": [
    "Christian Urech"
  ],
  "comment": "33 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "We look at algebraic embeddings of the Cremona group in $n$ variables $Cr_n(C)$ to the group of birational transformations $Bir(M)$ of an algebraic variety $M$. First we study geometrical properties of an example of an embedding of $Cr_2(C)$ into $Cr_5(C)$ that is due to Gizatullin. In a second part, we give a full classification of all algebraic embeddings of $Cr_2(C)$ into $Bir(M)$, where $dim(M)=3$, and generalize this result partially to algebraic embeddings of $Cr_n(C)$ into $Bir(M)$, where $dim(M)=n+1$, for arbitrary $n\\geq 2$. In particular, this yields a classification of all algebraic $PGL_{n+1}(C)$-actions on smooth projective varieties of dimension $n+1$ that can be extended to rational actions of $Cr_n(C)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-10T15:09:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14E07",
      "14L30",
      "32M05"
    ],
    "keywords": [
      "cremona group",
      "algebraic embeddings",
      "homomorphisms",
      "smooth projective varieties",
      "birational transformations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160303294U"
    }
  }
}