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Detaching embedded points. (English) Zbl 1250.14004

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.

MSC:

14B07 Deformations of singularities
14H10 Families, moduli of curves (algebraic)
14H50 Plane and space curves

Citations:

Zbl 1053.14035
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