## Infinitely many minimal models

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

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?

-

Calabi-Yau varieties have, by definition, (numerically) trivial canonical bundles so are their own unique minimal models. Perhaps you meant to ask something different? – ulrich Sep 15 2011 at 5:57
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.  – Artie Prendergast-Smith Sep 15 2011 at 13:55
Also, note that the OP is asking about marked minimal models, which introduces even more non-uniqueness into the picture! – Artie Prendergast-Smith Sep 15 2011 at 13:56
@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.  – ulrich Sep 15 2011 at 14:24
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.) – Artie Prendergast-Smith Sep 15 2011 at 14:33

## 2 Answers

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.

-

### You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.

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?

-
Hi John, you could ask the moderators to merge your two accounts. Or you could simply flag this answer for moderator attention. – Artie Prendergast-Smith Sep 15 2011 at 23:57

2
Dear John, I made a request on meta for your accounts to be merged. Regards, Matthew – Emerton Sep 16 2011 at 1:54

1
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 7, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9494990110397339, "perplexity_flag": "middle"}