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

We analyze the Gorenstein locus of the Hilbert scheme of d points on n i.e., the open subscheme parameterizing zero-dimensional Gorenstein subschemes of 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 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 n for d 4.

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