Equigenerated Gorenstein ideals of codimension three

• Lira, Dayane ^[1] ; Ramos, Zaqueu ^[2] ; Simis, Aron ^[3]
  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=?[x1,…,xn] over an infinite field ?, 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≥2, an ideal I⊂?[x1,…,xn] generated by a general set of r≥n+2 forms of degree d≥2 is Gorenstein if and only if d=2 and r=(n+12)−1. We prove the ‘only if’ implication of this conjecture when n=3. For arbitrary n≥2, we prove that if d=2 and r≥(n+2)(n+1)/6 then the ideal is Gorenstein if and only if r=(n+12)−1, which settles the ‘if’ assertion of the conjecture for n≤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 (ℓm1,…,ℓmn):f is equigenerated, where ℓ1,…,ℓn are independent linear forms and f is a form. We give a solution in some special cases.