GIT semistability of Hilbert points of Milnor algebras

TL;DR

This paper establishes a link between GIT semistability of homogeneous forms and the semistability of associated Hilbert points of Milnor algebras, providing new insights into the geometric invariant theory of these forms.

Contribution

It proves that GIT semistability of a form is equivalent to the semistability of its Milnor algebra's first non-trivial Hilbert point and shows the associated form is semistable for smooth forms, answering open questions.

Findings

GIT semistability of forms corresponds to semistability of Milnor algebra Hilbert points.

The morphism on GIT quotients is finite and injective on stable forms.

Associated forms of smooth forms are GIT semistable.

Abstract

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 first non-trivial Hilbert point of the associated Milnor algebra 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 from arXiv:1407.6838.