There are examples of elliptic fiber spaces over a two-dimensional base which have infinitely many relative minimal models (where two abstractly isomorphic models connected by flops are counted separately). The one I know is given by Reid and Kawamata and works by repeatedly flopping two rational curves in a singular fiber. In Matsuki's "Introduction to the Mori Program" he indicates (pg. 366) that this construction can be extended to a non-relative setting to yield a variety $X$ with infinitely many minimal models over Spec k. 


I haven't managed to make this extension or find it written down, so a couple questions: where can I find an explicit example of a variety with infinitely many minimal models (over Spec k)? What is the Kodaira dimension in this case? Is it possible to find a Calabi-Yau threefold with infinitely many minimal models?



Infinitely many minimal models
Dear John, 


One example can be found in [this paper by Fryers](http://arxiv.org/abs/math/0102055). This example is a Horrocks--Mumford quintic, in particular a Calabi--Yau threefold. He shows there are infinitely many marked minimal models, but they fall into only 8 (unmarked) isomorphism classes.


I think there is another class of examples in higher dimensions in [this recent paper of Oguiso](http://arxiv.org/abs/1107.5862). I must admit I haven't looked at the paper too closely, but if I remember correctly a talk I heard him give, there are indeed infinitely many marked models in this case. There is also a nice picture involving the geometry of Coxeter groups, but sadly that didn't seem to make it into the paper. 


