Elliptic Three-folds I: Ogg-Shafarevich Theory

1992/10/30 by Igor Dolgachev, Dolgachev, I., Mark Gross +1 · 2 citations
Mathematics · #Algebraic Geometry and Number Theory #Finite Group Theory Research #Advanced Differential Equations and Dynamical Systems

paper · pdf · doi:10.48550/arxiv.alg-geom/9210009

Abstract

We calculate the Tate-Shafarevich group of an elliptic three-fold f:X→ S when X and S are regular and f is flat, relating it to the Brauer group of X and S. We show that given certain hypotheses on f, the Tate-Shafarevich group has the interpretation of isomorphism classes of elliptic curves over the function field of S which have the same jacobian as the generic fibre of f, and for which there exists a relatively minimal model which has no multiple fibres. We use this to give examples of elliptic fibrations with isolated multiple fibres, and also to give a new counterexample to the Luroth problem in dimension three. This is a revised, hopefully improved, version with a few extra theorems and a few errors corrected.

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