×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] Artin, M. : Versal Deformations and Algebraic Stacks , Inv. Math. 27 (1974) 165-189. · Zbl 0317.14001
[2] Bingener, J. : Offenheit der Versalität in der analytischen Geometrie , Math. Z. 173 (1980) 241-281. · Zbl 0493.32020
[3] Bingener, J. : Darstellbarkeitskriterien für analytische Funktoren . Ann. Sci. École Norm. Sup. 13 (1980) 317-347. · Zbl 0454.32017
[4] Buchweitz, R. and Greuel, G. - M.: The Milnor number and deformations of complex curve singularities . Invent. Math. 58 (1980) 241-281. · Zbl 0458.32014
[5] Flenner, H. : Über Deformationen holomorpher Abbildungen , Habilitationsschrift, Osnabrück (1978).
[6] Flenner, H. : Ein Kriterium für die Offenheit der Versalität . Math. Z. 178 (1981) 449-473. · Zbl 0453.14002
[7] Greuel, G.-M. : A remark on the paper of A. Tannenbaum , Comp. Math. 51 (1984) 185-187. · Zbl 0556.14009
[8] Illusie, L. : Complex cotangent et deformations I , SLN 239, Springer Verlag (1971). · Zbl 0224.13014
[9] Koelman, R.J. : Over de Cusp, Leiden (1986).
[10] Laudal, A. : Formal Moduli of Algebraic Structures , SLN 754, Springer-Verlag (1979). · Zbl 0438.14007
[11] Luengo, I. : On the Existence of Complete Families of Projective Plane Curves, which are Obstructed . Preprint. · Zbl 0641.14012
[12] Palamodov, V.P. : Deformations of Complex Spaces , Russian Math. Surv. 31 (1976) 129-197. · Zbl 0347.32009
[13] Schlessinger, M. : Functors of Artin rings , Trans AMS 130 (1968) 208-222. · Zbl 0167.49503
[14] Schuster, W. and Vogt, A. : The Moduli of quotients of a compact complex space . Journ. f.r.u.a. Math. 364 (1986) 51-59. · Zbl 0567.32007
[15] Tannenbaum, A. : Families of algebraic curves with nodes . Comp. Math. 41 (1980) 107-126. · Zbl 0399.14018
[16] Tannenbaum, A. : On the classical characteristic linear series of plane curves with nodes and cuspidal points: two examples of Benjamino Segre . Comp. Math. 51 (1984) 169-183. · Zbl 0556.14008
[17] Van Straten, D. : Weakly Normal Surface Singularities and their Improvements , Thesis, Leiden 1987.
[18] Wahl, J. : Deformations of plane curves with nodes and cusps . Am. Journ. of Math. 96 (1974) 529-577. · Zbl 0299.14008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.