GIT semistability of Hilbert points of Milnor algebras

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• 作者：Maksym Fedorchuk
• 刊名：Mathematische Annalen
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：367
• 期：1-2
• 页码：441-460
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-1807
• 卷排序：367

文摘

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.