Let $S=K[x,y,z]$. Consider $f\in S_{2d}$ a ternary form of even degree and let $I_f$ be the associated Gorenstein ideal, i.e., all polynomials $G$ such that $G(\partial_x,\partial_y,\partial_z)f=0$. Assume that $\dim_K(S/I_f)_i=\min(\dim S_{i},\dim S_{2d-i})$ which is true for generic $f$. Is it true that $(I_f^2)_{2d+2}=S_{2d+2}$? I can prove it for $d=1$ and suspect it to be true for any $d>0$. In the book "Power Sums, Gorenstein Algebras, and Determinantal Loci" by Iarrobino and Kanev I was only able to find Thm.4.1C, which is about non-generic $f$.
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$\begingroup$ I think it might be true in more than $3$ variables, too. (I only computed a few examples, so I'm not sure, but it seems to be correct.) $\endgroup$– Zach TeitlerCommented Nov 16, 2019 at 6:21
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$\begingroup$ That would be very nice! $\endgroup$– HansCommented Nov 17, 2019 at 10:10
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1$\begingroup$ There is an easy weaker statement: $(I_f^k)_{k(d+1)} = S_{k(d+1)}$ for some $k$. To see this, first of all, you have the hypothesis that $S/I_f$ is what Iarrobino calls "compressed", and it implies that $(I_f)_d = 0$. So the minimum degree generators of $I_f$ are in degree $d+1$. Gorenstein symmetry implies the maximum degree generators of $I_f$ are also in degree $d+1$. So $I_f$ is generated purely in degree $d+1$. Next: $I_f$ is $\mathfrak{m}$-primary for the irrelevant (homogeneous maximal) ideal $\mathfrak{m}$. So some $\mathfrak{m}^s \subset I_f$, and it implies my weaker claim. $\endgroup$– Zach TeitlerCommented Sep 13, 2023 at 18:21
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