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On complex projective hypersurfaces

Published online by Cambridge University Press:  20 January 2009

J. W. Bruce
J. W. Bruce
Affiliation:
University of Liverpool
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In this paper we prove various results concerning monodromy groups associated with nonsingular complex projective hypersurfaces. Most of these results are already known but proofs are either unavailable or are algebraic and require a lot of machinery. The groups in question are those obtained from the second Lefschetz theorem (see (1)) applied to (a) the general Veronese variety, (b) a nonsingular projective hypersurface. By embedding the monodromy group of an extraordinary local isolated singularity (discovered by Libgober (8)) in these global monodromy groups we obtain necessary and sufficient conditions for the global groups to be finite. For case (a) we also obtain information on the structure of the dual to the Veronese variety which is of use when considering the monodromy group. The author gratefully acknowledges the financial support of the Stiftung Volkswagenwerk for a vist to the IHES during which this paper was written.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 24 , Issue 2 , June 1981 , pp. 91 - 97
Copyright
Copyright © Edinburgh Mathematical Society 1981

References

REFERENCES

(1)Andreotti, A. and Erankel, T., ‘The Lefschetz hyperplane theorems’, Global Analysis-Papers in Honour of K. Kodaira, Ed.Google Scholar
(2)Arnol'd, V. I., ‘Normal forms for functions in neighbourhoods of degenerate critical points’, Russian Math. Surveys 29 (2) (1974), 11–49.CrossRefGoogle Scholar
(3)Baker, H. F., Principles of geometry, volume VI, algebraic surfaces Cambridge University Press.Google Scholar
(4)Bruce, J. W., ‘A stratification of the space of cubic surfaces’, Math. Proc. Cambridge Philos, Soc. 87 (1980), 427–441.Google Scholar
(5)Gabrielov, A. M., ‘Dynkin diagrams for unimodal singularities’, Funk. Anal. Appl. 8 (1974), 1–6.Google Scholar
(6)Hamm, H. and Trang, Le Dung, ‘Un theoreme de Zariski du type de Lefschetz’, Ann. scient. EC. Norm. Sup. 6 (1973), 317–366.Google Scholar
(7)Lamotke, K., ‘Die homologie isolierter Singularitaten’, Math. Zeitschrift 143 (1975), 27–44.Google Scholar
(8)Libgober, A., ‘A geometrical procedure for killing the middle dimensional homology groups of algebraic hypersurfaces’, Proc. Amer. Math. Soc. 63 (1977), 198–202.Google Scholar
(9)Mather, J., ‘Generic projections’, Annals of Maths. 98 (1973), 226–245.CrossRefGoogle Scholar
(10)Milnor, J. W., ‘Singular points of complex hypersurfaces’ (Annals of Maths. Studies 61, Princeton University Press, 1968).Google Scholar
(11)Milnor, J. W. and Stasheff, J., ‘Characteristic classes’ (Annals of Maths. Studies 76 (Princeton University Press, 1974).Google Scholar
(12)Mumford, D., ‘Algebraic geometry I: projective varieties’ (Springer Verlag, 1976).Google Scholar
(13)Todd, J. A., ‘On the topology of certain algebraic three fold loci’, Proc. Edinburgh Math. Soc. 2 (4) III (1935), 175–184.Google Scholar