Git Semistability of Hilbert Points of Milnor Algebras

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 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.