Irreducibility of the Gorenstein loci of Hilbert schemes via ray families

Casnati, Gianfranco; Jelisiejew, Joachim; Notari, Roberto

doi:10.2140/ant.2015.9.1525

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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

DOI: 10.2140/ant.2015.9.1525

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 $d \leq 13$ 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 $d \geq 4$ .

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

Vol. 9, No. 7, 2015