{
  "id": "1902.05119",
  "version": "v1",
  "published": "2019-02-13T20:42:33.000Z",
  "updated": "2019-02-13T20:42:33.000Z",
  "title": "Overdetermined Systems of Equations on Toric, Spherical, and Other Algebraic Varieties",
  "authors": [
    "Leonid Monin"
  ],
  "comment": "16 pages, comments are welcome!",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $E_1,\\ldots,E_k$ be a collection of linear series on an algebraic variety $X$ over $\\mathbb{C}$. That is, $E_i\\subset H^0(X, \\mathcal{L}_i)$ is a finite dimensional subspace of the space of regular sections of line bundles $ \\mathcal{L}_i$. Such a collection is called overdetermined if the generic system \\[ s_1 = \\ldots = s_k = 0, \\] with $s_i\\in E_i$ does not have any roots on $X$. In this paper we study solvable systems which are given by an overdetermined collection of linear series. Generalizing the notion of a resultant hypersurface we define a consistency variety $R\\subset \\prod_{i=1}^k E_i$ as the closure of the set of all systems which have at least one common root and study general properties of zero sets $Z_{\\bf s}$ of a generic consistent system ${\\bf s}\\in R$. Then, in the case of equivariant linear series on spherical homogeneous spaces we provide a strategy for computing discrete invariants of such generic non-empty set $Z_{\\bf s}$. For equivariant linear series on the torus $(\\mathbb{C}^*)^n$ this strategy provides explicit calculations and generalizes the theory of Newton polyhedra.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-13T20:42:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "algebraic variety",
      "overdetermined systems",
      "equivariant linear series",
      "collection",
      "finite dimensional subspace"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}