Some elementary components of the Hilbert scheme of points (Q2409595)

Fix an algebraically closed field \(k\) of characteristic zero and let \(H^d (\mathbb A^n_k)\) denote the Hilbert scheme parametrizing zero dimensional closed subschemes of length \(d\). The \textit{principal component} of \(H^d (\mathbb A^n_k)\) consists of flat limits of \(d\) distinct points, but sometimes there are other components, the first examples being due to \textit{A. Iarrobino} [Invent. Math. 15, 72--77 (1972; Zbl 0227.14006)]. An \textit{elementary component} of \(H^d (\mathbb A^n_k)\) is one whose general member corresponds to a closed subscheme supported at a single point and is generically smooth. \textit{A. Iarrobino} and \textit{J. Emsalem} constructed an explicit example of an elementary component some 40 years ago [Compos. Math. 36, 145--188 (1978; Zbl 0393.14002)]. After describing their example carefully, the author uses the theory of border basis schemes due to \textit{M. Kreuzer} and \textit{L. Robbiano} [Collect. Math. 59, No. 3, 275--297 (2008; Zbl 1190.13022)]; J. Pure Appl. Algebra 215, No. 8, 2005--2018 (2011; Zbl 1216.13018)] to extend the Iarrobino-Ensalem construction, obtaining more examples of elementary components.