Irreducibility of the Gorenstein loci of Hilbert schemes via ray families

Abstract

We analyse the Gorenstein locus of the Hilbert scheme of $d$ points on ${\mathbb{P}}^n$ i.e. the open subscheme parameterising zero-dimensional Gorenstein subschemes of ${\mathbb{P}}^n$ 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 ${\mathbb{P}}^n$ for $d\ge 4$.

v4: final. v2: expanded proof of Theorems A and B. 33 pages, comments welcome!

References in corpus (2)

• Deformations of zero-dimensional schemes and applications
• On the rationality of Poincaré series of Gorenstein algebras via Macaulay's correspondence

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