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Equigenerated Gorenstein ideals of codimension three.
- Source :
- Collectanea Mathematica; Sep2023, Vol. 74 Issue 3, p567-593, 27p
- Publication Year :
- 2023
-
Abstract
- We focus on the structure of a homogeneous Gorenstein ideal I of codimension three in a standard polynomial ring R = k [ x 1 , ... , x n ] over an infinite field 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 ≥ 2 , an ideal I ⊂ k [ x 1 , ... , x n ] 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 + 1 2 - 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 + 1 2 - 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 (ℓ 1 m , ... , ℓ n m) : f is equigenerated, where ℓ 1 , ... , ℓ n are independent linear forms and f is a form. We give a solution in some special cases. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00100757
- Volume :
- 74
- Issue :
- 3
- Database :
- Complementary Index
- Journal :
- Collectanea Mathematica
- Publication Type :
- Academic Journal
- Accession number :
- 164899965
- Full Text :
- https://doi.org/10.1007/s13348-022-00365-6