## Families of varieties with prescribed singularities.(English)Zbl 0684.32015

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 $${\mathcal D}'_{X/S}$$ 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 $${\mathcal D}'_{X/S}$$. 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 $$H^ 1(C,{\mathcal N}'_{C/S})$$ 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.

### MSC:

 32G10 Deformations of submanifolds and subspaces 32Sxx Complex singularities 14D15 Formal methods and deformations in algebraic geometry
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### References:

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