## 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

### Keywords:

Hilbert scheme; embedded points; deformation; space curve

Zbl 1053.14035
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