Irreducibility of the Gorenstein loci of Hilbert schemes via ray families

2014/05/31 by Gianfranco Casnati, Joachim Jelisiejew, Roberto Notari
Mathematics · #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Degree (music) #Hilbert scheme #Irreducibility #Locus (genetics) #Mathematical analysis #Mathematics #Pure mathematics #Smoothness #Tensor decomposition and applications #math.AC #math.AG #msc:13H10 #msc:14C05 #msc:14D15

paper · pdf · doi:10.2140/ant.2015.9.1525

published as Algebra Number Theory 9 (2015), no. 7, 1525--1570 · v4: final. v2: expanded proof of Theorems A and B. 33 pages, comments welcome!

openalex publication_date 2015/09/22 · arxiv created 2015/11/26 · arxiv updated 2015/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We analyze the Gorenstein locus of the Hilbert scheme of d points on Pⁿ i.e., the
\nopen subscheme parameterizing zero-dimensional Gorenstein subschemes of Pⁿ
\nof degree d. We give new sufficient criteria for smoothability and smoothness of
\npoints of the Gorenstein locus. In particular we prove that this locus is irreducible
\nwhen d <= 13 and find its components when d = 14.
\nThe proof is relatively self-contained and it does not rely on a computer algebra
\nsystem. As a by-product, we give equations of the fourth secant variety to the
\nd-th Veronese reembedding of Pⁿ for d >= 4.

Citations

• Algebraic Families on an Algebraic Surface
• On the rationality of Poincaré series of Gorenstein algebras via Macaulay's correspondence
• Algebraic Geometry
• Commutative Algebra