Abstract
We study GIT semistability of Hilbert points of Milnor algebras of homogeneous forms. Our first result is that a homogeneous form F in n variables is GIT semistable with respect to the natural \({{\mathrm{SL}}}(n)\)-action if and only if the gradient point of F, which is the first non-trivial Hilbert point of the Milnor algebra of F, is semistable. We also prove that the induced morphism on the GIT quotients is finite, and injective on the locus of stable forms. Our second result is that the associated form of F, also known as the Macaulay inverse system of the Milnor algebra of F, and which is apolar to the last non-trivial Hilbert point of the Milnor algebra, is GIT semistable whenever F is a smooth form. These two results answer questions of Alper and Isaev.
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Notes
Clearly, the sum of \(\lambda \)-weights of all degree \(n(d-1)\) monomials is 0.
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Acknowledgments
I am grateful to Jarod Alper for introducing me to the problem of GIT stability of Hilbert points of Milnor algebras. I had many illuminating discussions on the subject with Jarod Alper and Alexander Isaev. I would also like to thank Alexander Isaev and Ian Morrison for comments on earlier drafts. This research was partially supported by the NSF Grant DMS-1259226 and a Sloan Research Fellowship. The first part of the paper was completed when the author visited Columbia University in June 2015. The second part was completed when the author visited the Max Planck Institute for Mathematics in Bonn in October 2015. I thank these institutions for their hospitality.
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Fedorchuk, M. GIT semistability of Hilbert points of Milnor algebras. Math. Ann. 367, 441–460 (2017). https://doi.org/10.1007/s00208-016-1377-2
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DOI: https://doi.org/10.1007/s00208-016-1377-2