Elliptic Three-folds I: Ogg-Shafarevich Theory

TL;DR

This paper explores the Tate-Shafarevich group of elliptic three-folds, relating it to the Brauer group, and provides new insights and examples, including counterexamples to the Luroth problem in three dimensions.

Contribution

It calculates the Tate-Shafarevich group for elliptic three-folds and relates it to the Brauer group, offering new examples and counterexamples in algebraic geometry.

Findings

Tate-Shafarevich group relates to the Brauer group of the three-fold.

Provides examples of elliptic fibrations with isolated multiple fibres.

Offers a new counterexample to the Luroth problem in dimension three.

Abstract

We calculate the Tate-Shafarevich group of an elliptic three-fold $f:X\rightarrow S$ 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.