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Distribution of critical values of miniversal deformations of parabolic singularities. (English) Zbl 0578.32037
In this paper the critical value mapping is considered, which associates with any polynomial from underdiagonal miniversal deformation of the family of parabolic singularities the set of its critical values. It is shown that the restriction of this mapping to the subset of polynomials with \(k\) different critical values, \(k\geq 2\), is a covering of the space of unordered \(k\)-tuples of different complex numbers. In particular, it is proved that the connected components of such subsets are \(K[\pi,1]\) spaces.
Reviewer: Piotr Jaworski

MSC:
32S30 Deformations of complex singularities; vanishing cycles
32S05 Local complex singularities
32Sxx Complex singularities
14E20 Coverings in algebraic geometry
14B05 Singularities in algebraic geometry
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References:
[1] [B-G] Bruce, J.W., Giblin, P.J.: A stratification of the space of plane quartic curves. Proc. Lond. Math. Soc.42, 270?298 (1981) · Zbl 0446.14009
[2] [B-W] Bruce, J.W., Wall, C.T.C.: On the classification of cubic surfaces. J. Lond. Math. Soc.19, 245?256 (1979) · Zbl 0406.14020
[3] [B] Broughton, S.A.: On the topology of polynomial hypersurfaces. In: Singularities. Proc. of Symposia in pare math.40/1, 167?178 (1982) · Zbl 0526.14010
[4] [H] Hartshorne, R.: Algebraic geometry. Berlin Heidelberg New York: Springer 1977 · Zbl 0367.14001
[5] [K] Kouchnirenko, A.G.: Polyedres de Newton et nombers de Milnor. Invent. Math.32, 1?31 (1976) · Zbl 0328.32007
[6] [M] Milnor, J.: Singular points of complex hypersurfaces. Princeton: University Press 1968 · Zbl 0184.48405
[7] [S] Saito, K.: Einfach-elliptische Singularit√§ten. Invent. Math.23, 289?325 (1974) · Zbl 0296.14019
[8] [W-1] Wall, C.T.C.: Note on the invariant of plane cubics. Math. Proc. Cambr. Philos. Soc.85, 403?406 (1979) · Zbl 0404.14002
[9] [W-2] Wall, C.T.C.: Affine cubic functions IV. Functions onC 3, nonsingular at infinity. Phil. Trans. R. Soc. Lond. Ser. A302, 415?455, no 1470 (1981) · Zbl 0462.14008
[10] [ODO] ??????? ?.?, ???????? ?.?., ??????-???? ??.: ??????????? ???????????????? ???????????. ?????, ?????? 1982/1984
[11] [J] ??????? ?.: ?????????? ?????????????? ???????????? ?? ????? ??????. ??????? ????. ??-??. Cep. I. (to appear)
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