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, l… Show more
“…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].…”
We use generalised Białynicki-Birula decomposition, apolarity and obstruction theories to prove non-reducedness of the Hilbert scheme of 13 points on A 6 . Our argument doesn't involve computer calculations and gives an example of a fractal-like structure on this Hilbert scheme.
CONTENTS1. Introduction 1 2. Preliminaries 3 2.1. Hilbert schemes 4 2.2. Groups acting on H 5 2.3. Białynicki-Birula decomposition 7 2.4. Obstruction theories 8 2.5. Barycenter 13 2.6. Macaulay duality 14 3. The main example 17 3.1. The analysis of the point [I] 18 3.2. Zero-dimensionality of V 26 3.3. Fractal family and its flatness 28 3.4. Complete description of the fiber V 31 Acknowledgements 35 References 35
“…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 .…”
Section: Preliminaries On Finite Schemesmentioning
confidence: 99%
“…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.…”
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