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Vanishing cycles in a pencil of hyperplane sections of a non-singular quasi-projective variety. (English) Zbl 0851.14003

We give an inductive way to obtain the vanishing cycles which occur in the “second Lefschetz theorem” about pencils of hyperplane sections of a non-singular projective variety. In fact we do this for the more general case of a non-singular quasi-projective variety. As a corollary, we show that the Lefschetz hyperplane section theorem for a non-singular quasi-projective variety can be made more precise in the middle dimension by a statement which recalls the theorem of von Kampen on the fundamental group of the complement of a plane projective curve.

MSC:

14C25 Algebraic cycles
14C21 Pencils, nets, webs in algebraic geometry
14J70 Hypersurfaces and algebraic geometry
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