Elliptic Three-folds I: Ogg-Shafarevich Theory

I. Dolgachev; M. Gross

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Elliptic Three-folds I: Ogg-Shafarevich Theory

I. Dolgachev·M. Gross

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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.

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BibTeX RIS

I. Dolgachev, M. Gross. 1993-04-16. Elliptic Three-folds I: Ogg-Shafarevich Theory. https://arxiv.org/abs/alg-geom/9210009

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