Rocky Mountain Journal of Mathematics

Some elementary components of the Hilbert scheme of points

Mark E. Huibregtse

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Let $K$ be an algebraically closed field of characteristic~$0$, and let $H^{\mu }_{\mathbb {A}^n_{K }}$ denote the Hilbert scheme of $\mu $ points of $\mathbb {A}^n_{K}$. An \textit {elementary component} $E$ of $H^{\mu }_{\mathbb {A}^n_{K }}$ is an irreducible component such that every $K$-point $[I]\in E$ represents a length-$\mu $ closed subscheme $Spec (K [x_1,\ldots ,x_n]/I)\subseteq \mathbb {A}^n_{K}$ that is supported at one point. Iarrobino and Emsalem gave the first explicit examples (with $\mu > 1$) of elementary components \cite {Iarrob-Emsalem}; in their examples, the ideals $I$ were homogeneous (up to a change of coordinates corresponding to a translation of $\mathbb {A}^n_{K}$). We generalize their construction to obtain new examples of elementary components.

Article information

Rocky Mountain J. Math., Volume 47, Number 4 (2017), 1169-1225.

First available in Project Euclid: 6 August 2017

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Zentralblatt MATH identifier

Primary: 14C05: Parametrization (Chow and Hilbert schemes)

Generic algebra small tangent space Hilbert scheme of points elementary component


Huibregtse, Mark E. Some elementary components of the Hilbert scheme of points. Rocky Mountain J. Math. 47 (2017), no. 4, 1169--1225. doi:10.1216/RMJ-2017-47-4-1169.

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