{
  "id": "2303.09582",
  "version": "v1",
  "published": "2023-03-16T18:02:58.000Z",
  "updated": "2023-03-16T18:02:58.000Z",
  "title": "Monomial projections of Veronese varieties: new results and conjectures",
  "authors": [
    "Liena Colarte-Gómez",
    "Rosa M. Miró-Roig",
    "Lisa Nicklasson"
  ],
  "comment": "To appear in Journal of Algebra",
  "categories": [
    "math.AG",
    "math.AC"
  ],
  "abstract": "In this paper, we consider the homogeneous coordinate rings $A(Y_{n,d}) \\cong \\mathbb{K}[\\Omega_{n,d}]$ of monomial projections $Y_{n,d}$ of Veronese varieties parameterized by subsets $\\Omega_{n,d}$ of monomials of degree $d$ in $n+1$ variables where: (1) $\\Omega_{n,d}$ contains all monomials supported in at most $s$ variables and, (2) $\\Omega_{n,d}$ is a set of monomial invariants of a finite diagonal abelian group $G \\subset GL(n+1,\\mathbb{K})$ of order $d$. Our goal is to study when $\\mathbb{K}[\\Omega_{n,d}]$ is a quadratic algebra and, if so, when $\\mathbb{K}[\\Omega_{n,d}]$ is Koszul or G-quadratic. For the family (1), we prove that $\\mathbb{K}[\\Omega_{n,d}]$ is quadratic when $s \\ge \\lceil \\frac{n+2}{2} \\rceil$. For the family (2), we completely characterize when $\\mathbb{K}[\\Omega_{2,d}]$ is quadratic in terms of the group $G \\subset GL(3,\\mathbb{K})$, and we prove that $\\mathbb{K}[\\Omega_{2,d}]$ is quadratic if and only if it is Koszul. We also provide large families of examples where $\\mathbb{K}[\\Omega_{n,d}]$ is G-quadratic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-16T18:02:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "veronese varieties",
      "monomial projections",
      "conjectures",
      "finite diagonal abelian group",
      "quadratic algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}