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- W3029412099 abstract "We focus on the structure of a homogeneous Gorenstein ideal I of codimension three in a standard polynomial ring $$R={mathbbm{k}},[x_{1},ldots ,x_{n}]$$ over an infinite field $${mathbbm{k}}$$ , 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 $$nge 2$$ , an ideal $$Isubset {mathbbm{k}}[x_1,ldots ,x_n]$$ generated by a general set of $$rge n+2$$ forms of degree $$dge 2$$ 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 $$nge 2$$ , we prove that if $$d=2$$ and $$rge (n+2)(n+1)/6$$ then the ideal is Gorenstein if and only if $$r={{n+1}atopwithdelims ()2}-1$$ , which settles the ‘if’ assertion of the conjecture for $$nle 5$$ . 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." @default.
- W3029412099 created "2020-06-05" @default.
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- W3029412099 date "2022-06-03" @default.
- W3029412099 modified "2023-10-18" @default.
- W3029412099 title "Equigenerated Gorenstein ideals of codimension three" @default.
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- W3029412099 doi "https://doi.org/10.1007/s13348-022-00365-6" @default.
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