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

TL;DR

This paper proves the existence of good hyperplane sections and Lefschetz pencils over discrete valuation rings, and applies these results to establish an isomorphism of the reciprocity map in higher class field theory for varieties with good or nearly good reduction.

Contribution

It introduces new geometric techniques for schemes over discrete valuation rings and applies them to prove a key isomorphism in higher class field theory.

Findings

Existence of good hyperplane sections for schemes over DVRs with good or semistable reduction.

Existence of good Lefschetz pencils for schemes with good or ordinary quadratic reduction.

Proves the reciprocity map is an isomorphism after profinite completion for certain varieties.

Abstract

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