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Zbl 0684.32015
Greuel, Gert-Martin; Karras, Ulrich
Families of varieties with prescribed singularities. (English)
[J] Compos. Math. 69, No.1, 83-110 (1989). ISSN 0010-437X; ISSN 1570-5846/e

In the sections 1-3 of the paper the authors consider deformations of a holomorphic map f: $X\to S$ between complex spaces, with fixed base S such that the induced deformation of X is locally trivial in each point of X. If ${\cal D}'\sb{X/S}$ is the associated functor of isomorphism classes of such deformations, then it is shown that for compact X with isolated singularities there exists a convergent miniversal locally trivial deformation space and that the opennes of versality property holds for ${\cal D}'\sb{X/S}$. In the sections 4-6 of the paper the authors apply this to families of reduced curves, in particular to embedded deformations of curves C lying on a smooth surface S. It is proved the existence of a convergent versal deformation space which is algebraic, if the curve has only simple singularities. For an arbitrary smooth surface S containing C there are given sufficient conditions for $H\sp 1(C,{\cal N}'\sb{C/S})$ to be zero, which implies the independence of conditions imposed by the singularities. These sufficient conditions are given in terms of the genera, intersection numbers and Tjurina numbers of the irreducible components of C and are very easy to compute.
[Vasile Brînzănescu]

MSC 2000:: *32G10 Deformations of submanifolds
32Sxx Singularities
14D15 (Formal) deformations

Keywords: families of varieties; isolated singularities; embedded deformations

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