#### 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