Classifying local Artinian Gorenstein algebras

Abstract

The classification of local Artinian Gorenstein algebras is equivalent to the study of orbits of a certain non-reductive group action on a polynomial ring. We give an explicit formula for the orbits and their tangent spaces. We apply our technique to analyse when an algebra is isomorphic to its associated graded algebra. We classify algebras with Hilbert function (1, 3, 3, 3, 1), obtaining finitely many isomorphism types, and those with Hilbert function (1, 2, 2, 2, 1, 1, 1). We consider fields of arbitrary, large enough, characteristic.

Artinian level algebras of socle degree 4

“…To prove that a local K-algebra is not canonically graded is in general a very difficult task. See also [Jel16] for interesting discussions.…”

Non-reducedness of the Hilbert schemes of few points

“…In order to find hedgehog points on the barycenter scheme we will use the construction of apolar algebras. More thorough introduction of this topic can be found in [Jel17].…”

VSPs of cubic fourfolds and the Gorenstein locus of the Hilbert scheme of 14 points on A6

“…If R is Gorenstein then the socle degree of R is the largest d such that H(d) > 0. Let R be Gorenstein of socle degree d. We will use the following properties of the Hilbert function (see [BB14,Iar94]): IK99,Jel16] for general facts about Macaulay's inverse systems. In this section we work over an arbitrary algebraically closed field k and with a finite dimensional k-vector space V with n = dim V .…”

“…The freedom of choice of F is well understood, see for example [Jel16]. If the algebra in question is naturally graded, we may take F to be homogeneous.…”