Small elementary components of Hilbert schemes of points

Abstract

AbstractWe answer an open problem posed by Iarrobino,Hilbert scheme of points: Overview of last ten years. Proceedings of Symposia in Pure Mathematics, 46 (American Mathematical Society, Providence, RI, 1987), 297–320: Is there a component of the punctual Hilbert scheme [Grothendieck, Techniques de construction et théorèmes d'existence en géométrie algébrique. IV. Les schémas de Hilbert',in Séminaire Bourbaki, 6 (Societe Mathematique de France, Paris, 1995), 221, 249–276]$\operatorname {\mathrm {Hilb}}^d({\mathscr {O}}_{\mathbb {A}^n,p})$with dimension less than$(n-1)(d-1)$? For each$n\geq 4$, we construct an infinite class of elementary components in$\operatorname {\mathrm {Hilb}}^d(\mathbb {A}^n)$producing such examples. Our techniques also allow us to construct an explicit example of a local Artinian ring [Iarrobino and Kanev,Power sums, Gorenstein algebras, and determinantal loci(Springer-Verlag, Berlin, 1999), 221–226] of the formwith trivial negative tangents, vanishing nonnegative obstruction space, and socle-dimension$2$.