Irreducibility of the Gorenstein loci of Hilbert schemes via ray families

By G. Casnati, Joachim Jelisiejew, Roberto Notari. 2014

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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 $d$ 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 $d\ge4$

Subfields: Algebra and Number Theory, Computational Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3a93eda8c1f6f2950fcbe45b54cefad33591748e