Chen, Dawei; Nollet, Scott Detaching embedded points. (English) Zbl 1250.14004 Algebra Number Theory 6, No. 4, 731-756 (2012). Let \(Y \subset \mathbb{P}^N\) be a closed subscheme containing zero-dimensional embedded points and let \(X\) be the closed scheme obtained by removing the embedded points. The main question of the paper is: when does \(Y\) belong to a flat irreducible family having \(X\) union isolated points as its general member?The authors are able to give the following interesting answer: Suppose that the multiplicities of the embedded points are at most 3 and that \(X\) is locally a complete intersection of codimension 2, then \(Y\) is a flat specialization of \(X\) union isolated points. This result is optimal for the size of the multiplicity and the codimension, and also with respect to being a local complete intersection. Using S. Nollet and E. Schlesinger [Compos. Math. 139, No. 2, 169–196 (2003; Zbl 1053.14035)] the authors give examples of irreducible components of the Hilbert scheme \({\text{ H}}(d,g):={\text{ Hilb}}^{dz+1-g}(\mathbb{P}^3)\) of one-dimensional schemes of degree \(d=4\) and arithmetic genus \(g\) whose general point is a curve with an embedded point.Using their theorems they show several results for the Hilbert scheme \({\text{ H}}(d,g)\) of high genus, e.g. that \({\text{ H}}(d,g)\) is irreducible for \(d \geq 6\) and \(g > -4 +(d-1)(d-2)/2\), and also smooth in the case \(g = -1 +(d-1)(d-2)/2\). Finally they prove that \({\text{ H}}(4,0)\) consists of four irreducible components and they describe the general curves. Reviewer: Jan O. Kleppe (Oslo) Cited in 3 ReviewsCited in 6 Documents MSC: 14B07 Deformations of singularities 14H10 Families, moduli of curves (algebraic) 14H50 Plane and space curves Keywords:Hilbert scheme; embedded points; deformation; space curve Citations:Zbl 1053.14035 PDFBibTeX XMLCite \textit{D. Chen} and \textit{S. Nollet}, Algebra Number Theory 6, No. 4, 731--756 (2012; Zbl 1250.14004) Full Text: DOI arXiv Link