Non-reducedness of the Hilbert schemes of few points

Abstract

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

Elementary components of Hilbert schemes of points

“…Neither it follows from subsequent paper of Szachniewicz [Sza21]. In contrast, in the paper [JŠ22] the authors show that Quot 8 (A 4 ) has a generically nonreduced component.…”

“…For important singularity types, such as nonreduced ones, sharper bounds on dim X are known. Szachniewicz [Sza21] proved that Hilb d (A 6 ) is nonreduced for every d ≥ 13; it has an embedded component. See [Erm12,Sch23] for some results in similar direction on fixed loci.…”

On the Hilbert function of Artinian local complete intersections of codimension three

“…Moreover, the feeling is that these degree −1 tangent directions have no geometric "meaning", so for most cases they should not extend to higher order. If this is the case, then an argument similar to [130] proves nonreducedness.…”

Hilbert schemes of points and applications

“…Are there infinitely many elementary components of Hilb r (A 3 )? 1 Examples found in [Jel20] and for 13 points on A 6 , in [Sza21]. 2 The Gröbner fan does not work, see [Jel19].…”