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We show the existence of good hyperplane sections for schemes over discretevaluation rings with good or (quasi) semistable reduction, and the existence ofgood Lefschetz pencils for schemes with good reduction or ordinary quadraticreduction. As an application we prove that the reciprocity map introduced forsmooth projective varieties over local fields by Bloch, Kato and Saito is anisomorphism after profinite completion, if the variety has good reduction or'almost good' reduction.

Bertini theorems and Lefschetz pencils over discrete valuation rings, with applications to higher class field theory