Bertini theorems and Lefschetz pencils over discrete valuation rings, with applications to higher class field theory (Q2913220): Difference between revisions

Let \(X\subset {\mathbb P} ^{N}_{L}\) be a smooth quasi-projective variety over a field \(L\). A subscheme \(Z\subset {\mathbb P} ^{N}_{L}\) intersects \(X\) transversally if \(X\cdot Z=X{\times}_{{\mathbb P}^{N}}Z\) is smooth and of pure codimension \(\mathrm{codim}_{{\mathbb P}^{N}}(Z)\) in \(X.\) The Bertini theorem asserts that for infinite \(L\), there exists an \(L\)-rational hyperplane that intersects \(X\) transversally. Good families of hyperplane sections - Lefschetz pencils - proved to be useful in algebraic geometry. However, for arithmetic questions it is natural to consider models over Dedekind domains or discrete valuation rings (for local questions). The authors extend the classical constructions mentioned above to the case of schemes over a discrete valuation ring \(A.\) They show the existence of good hyperplane sections for schemes over \(A\) with good or (quasi-) semi- stable reduction and the existence of (good) Lefschetz pencils for schemes with good or ordinary quadratic reduction. As an applicalion the authors prove a nice result concerning the reciprocity map introduced for smooth projective varieties by Bloch, Kato and Saito.