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Equigenerated Gorenstein ideals of codimension three

  • Lira, Dayane [1] ; Ramos, Zaqueu [2] ; Simis, Aron [3] Árbol académico
    1. [1] Universidade Federal da Paraíba

      Universidade Federal da Paraíba

      Brasil

    2. [2] Universidade Federal de Sergipe

      Universidade Federal de Sergipe

      Brasil

    3. [3] Universidade Federal de Pernambuco

      Universidade Federal de Pernambuco

      Brasil

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  • Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 74, Fasc. 3, 2023, págs. 567-593
  • Idioma: inglés
  • DOI: 10.1007/s13348-022-00365-6
  • Texto completo no disponible (Saber más ...)
  • Resumen

    • We focus on the structure of a homogeneous Gorenstein ideal I of codimension three in a standard polynomial ring R = \mathbb{k} [x_1, \ldots, x_n] over an infinite field \mathbb{k}, assuming that I is generated in a fixed degree d. For such an ideal I, there is a simple formula relating this degree, the minimal number of generators of I, and the degree of the entries of the associated skew-symmetric matrix. We give an elementary characteristic-free argument to the effect that, for any such data linked by this formula, there exists a Gorenstein ideal I of codimension three satisfying them. We conjecture that, for arbitrary n\ge 2 , an ideal I \subseteq \mathbb{k}[x_1, \ldots, x_n] generated by a general set of r\ge n+2 forms of degree d\ge 2 is Gorenstein if and only if d=2 and r= {{n+1}\atopwithdelims ()2}-1. We prove the ‘only if’ implication of this conjecture when n=3. For arbitrary n\ge 2, we prove that if d=2 and r\ge (n+2)(n+1)/6 then the ideal is Gorenstein if and only if r={{n+1}\atopwithdelims ()2}-1, which settles the ‘if’ assertion of the conjecture for n\le 5. We also elaborate around one of the questions of Fröberg–Lundqvist. In a different direction, we show a connection between the Macaulay inverse and the so-called Newton dual, a matter so far not brought out to our knowledge. Finally, we consider the question as to when the link (\ell _1^m,\ldots ,\ell _n^m):\mathfrak{f} is equigenerated, where \ell _1,\ldots ,\ell _n are independent linear forms and \mathfrak{f} is a form. We give a solution in some special cases.

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