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?

  • $\begingroup$ Calabi-Yau varieties have, by definition, (numerically) trivial canonical bundles so are their own unique minimal models. Perhaps you meant to ask something different? $\endgroup$
    – naf
    Commented Sep 15, 2011 at 5:57
  • $\begingroup$ Dear ulrich, I'm not sure I understand your comment. (Terminal) Calabi--Yaus are certainly their own minimal models, but there's no reason they should be unique: if $f:X \dashrightarrow X'$ is a birational map of Calabi--Yaus, then each one is a minimal model of the other, but they need not be isomorphic. $\endgroup$
    – user5117
    Commented Sep 15, 2011 at 13:55
  • $\begingroup$ Also, note that the OP is asking about marked minimal models, which introduces even more non-uniqueness into the picture! $\endgroup$
    – user5117
    Commented Sep 15, 2011 at 13:56
  • $\begingroup$ @Artie: By minimal model of $X$ I assumed one means a variety $Y$ that is the end result of running the MMP on $X$. If $X$ is Calabi-Yau then the canonical bundle is nef so the MMP ends at $X$ itself. But having read the question again I agree that this is probably not the definition the OP has in mind. $\endgroup$
    – naf
    Commented Sep 15, 2011 at 14:24
  • $\begingroup$ Dear ulrich: you're right, people aren't always very careful to say exactly what they mean by "minimal model". (Here by "people" I mean "authors", not the OP.) $\endgroup$
    – user5117
    Commented Sep 15, 2011 at 14:33

1 Answer 1


Dear John,

One example can be found in this paper by Fryers. 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. 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.

  • $\begingroup$ The references are exactly what I was looking for. Thanks, Artie! I asked the question without logging in and I'm not sure how to accept this answer (or post this as a comment on it instead of an answer in its own right). Is there anything to be done? $\endgroup$
    – John L.
    Commented Sep 15, 2011 at 23:25

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