{
  "id": "2310.10361",
  "version": "v1",
  "published": "2023-10-16T12:56:55.000Z",
  "updated": "2023-10-16T12:56:55.000Z",
  "title": "Linear system of hypersurfaces passing through a Galois orbit",
  "authors": [
    "Shamil Asgarli",
    "Dragos Ghioca",
    "Zinovy Reichstein"
  ],
  "comment": "12 pages",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "Let $d$ and $n$ be positive integers, and $E/F$ be a separable field extension of degree $m=\\binom{n+d}{n}$. We show that if $|F| > 2$, then there exists a point $P\\in \\mathbb{P}^n(E)$ which does not lie on any degree $d$ hypersurface defined over $F$. In other words, the $m$ Galois conjugates of $P$ impose independent conditions on the $m$-dimensional $F$-vector space of degree $d$ forms in $x_0, x_1, \\ldots, x_n$. As an application, we determine the maximal dimension of an $F$-linear system $\\mathcal{L}$ of hypersurfaces such that every $F$-member of $\\mathcal{L}$ is irreducible over $F$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-16T12:56:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14N05",
      "14J70",
      "14G15"
    ],
    "keywords": [
      "linear system",
      "galois orbit",
      "hypersurfaces passing",
      "impose independent conditions",
      "separable field extension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}