{
  "id": "math/0410309",
  "version": "v1",
  "published": "2004-10-13T06:04:16.000Z",
  "updated": "2004-10-13T06:04:16.000Z",
  "title": "Noncomplete embeddings of rational surfaces",
  "authors": [
    "Euisung Park"
  ],
  "comment": "8 pages",
  "categories": [
    "math.AG",
    "math.AC"
  ],
  "abstract": "In this paper, we study the Castelnuovo-Mumford regularity of nonlinearly normal embedding of rational surfaces. Let $X$ be a rational surface and let $L \\in {Pic}X$ be a very ample line bundle. For a very ample subsystem $V \\subset H^0 (X,L)$ of codimension $t \\geq 1$, if $X \\hookrightarrow \\P (V)$ satisfies Property $N^S_1$, then ${Reg} (X) \\leq t+2$\\cite{KP}. Thus we investigate Property $N^S_1$ of noncomplete linear systems on X. And our main result is about a condition of the position of $V$ in $H^0 (X,L)$ such that $X \\hookrightarrow \\P (V)$ satisfies Property $N^S_1$. Indeed it is related to the geometry of a smooth rational curve of $X$. Also we apply our result to $\\P^2$ and Hirzebruch surfaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-10-13T06:04:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14N05",
      "14J26",
      "16E05"
    ],
    "keywords": [
      "rational surface",
      "noncomplete embeddings",
      "satisfies property",
      "smooth rational curve",
      "ample line bundle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....10309P"
    }
  }
}