标题： 椭圆三重体 I：奥格-沙法列维奇理论 显示英文标题

标题： Elliptic Three-folds I: Ogg-Shafarevich Theory

摘要： 我们计算椭圆三重的Tate-Shafarevich群$f:X\rightarrow S$当$X$和$S$是正则的且$f$是平坦的，将其与$X$和$S$的Brauer群相关联。 我们证明，给定关于$f$的某些假设，Tate-Shafarevich 群可以解释为在$S$的函数域上的椭圆曲线的同构类，这些椭圆曲线具有与$f$的一般纤维相同的雅可比，并且存在一个没有多重纤维的相对极小模型。 我们利用这一点来给出具有孤立多重纤维的椭圆纤维化的例子，同时也给出了三维空间中对Luroth问题的新反例。 这是经过修改的版本，希望有所改进，增加了一些额外的定理并修正了一些错误。 显示英文摘要

摘要： 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.