Vol. 9, No. 7, 2015

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Irreducibility of the Gorenstein loci of Hilbert schemes via ray families

Gianfranco Casnati, Joachim Jelisiejew and Roberto Notari

Vol. 9 (2015), No. 7, 1525–1570
Abstract

We analyze the Gorenstein locus of the Hilbert scheme of $d$ points on ${ℙ}^{n}$ i.e., the open subscheme parameterizing zero-dimensional Gorenstein subschemes of ${ℙ}^{n}$ of degree $d$. We give new sufficient criteria for smoothability and smoothness of points of the Gorenstein locus. In particular we prove that this locus is irreducible when $d\le 13$ and find its components when $d=14$.

The proof is relatively self-contained and it does not rely on a computer algebra system. As a by-product, we give equations of the fourth secant variety to the $d$-th Veronese reembedding of ${ℙ}^{n}$ for $d\ge 4$.

Keywords
Hilbert scheme of points, smoothability, Gorenstein algebra, secant variety
Mathematical Subject Classification 2010
Primary: 14C05
Secondary: 13H10, 14D15
Milestones
Received: 13 September 2014
Revised: 18 April 2015
Accepted: 17 June 2015
Published: 22 September 2015
Authors
 Gianfranco Casnati Dipartimento di Scienze Matematiche Politecnico di Torino corso Duca degli Abruzzi 24 I-10129 Torino Italy Joachim Jelisiejew Faculty of Mathematics, Informatics, and Mechanics University of Warsaw Banacha 2 02-097 Warsaw Poland Roberto Notari Dipartimento di Matematica “Francesco Brioschi” Politecnico di Milano via Bonardi 9 I-20133 Milano Italy