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Mori Flips, Cluster Algebras and Diptych Varieties Without Unprojection

Published online by Cambridge University Press:  25 October 2022

By
Tom Ducat
Edited by
Hamid Abban ,
Gavin Brown ,
Alexander Kasprzyk  and
Shigefumi Mori
Hamid Abban
Affiliation:
Loughborough University
Gavin Brown
Affiliation:
University of Warwick
Alexander Kasprzyk
Affiliation:
University of Nottingham
Shigefumi Mori
Affiliation:
Kyoto University, Japan
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Summary

We give a survey on the connections between terminal 3-fold flips and cluster algebras. In particular we observe that Mori’s algorithm for generating the relations defining a type k2A flipping neighbourhood is a form of generalised cluster algebra mutation. We then use the Laurent phenomenon for this cluster algebra structure to give an alternative proof of the existence of Brown and Reid’s diptych varieties.

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Chapter
Information
Recent Developments in Algebraic Geometry
To Miles Reid for his 70th Birthday
 , pp. 116 - 149
Publisher: Cambridge University Press
Print publication year: 2022

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References

Berenstein, Arkady, Sergey Fomin, and Andrei Zelevinsky. Cluster algebras. III. Upper bounds and double Bruhat cells. Duke Math. J., 126(l):l-52, 2005.CrossRefGoogle Scholar
Brown, Gavin, and Miles Reid. Diptych varieties. III: Redundant generators. In preparation, 10pp.Google Scholar
Brown, Gavin, and Miles Reid. Diptych varieties, I. Proc. Lond. Math. Soc. (3), 107(6):1353–1394, 2013.CrossRefGoogle Scholar
Brown, Gavin and Miles Reid. Diptych varieties. II: Apolar varieties. Higher dimensional algebraic geometry—in honour of Professor Yujiro Kawamata's sixtieth birthday, volume 74 of Adv. Stud. Pure Math., pages 41–72. Math. Soc. Japan, Tokyo, 2017.Google Scholar
Ducat. Mori extractions from singular curves in a smooth 3-fold. PhD thesis, Thomas, 2015. 126pp.Google Scholar
Gross, Mark, Paul Hacking, and Sean Keel. Mirror symmetry for log Calabi-Yau surfaces I. Publ. Math. Inst. Hautes Etudes Sci., 122:65–168, 2015.Google Scholar
Hacking, Paul, Jenia Tevelev, and Giancarlo Urzua. Flipping surfaces. J. Algebraic Geom., 26(2):279–345, 2017.Google Scholar
Kollár, János, and Shigefumi Mori. Classification of three-dimensional flips. J. Amer. Math. Soc., 5(3):533–703, 1992.CrossRefGoogle Scholar
P. Matherne, Jacob, and Greg Muller. Computing upper cluster algebras. Int. Math. Res. Not. IMRN, (11):3121–3149, 2015.Google Scholar
Mori, Shigefumi. On semistable extremal neighborhoods. In Higher dimensional birational geometry (Kyoto, 1997), volume 35 of Adv. Stud. Pure Math., pages 157–184. Math. Soc. Japan, Tokyo, 2002.Google Scholar
Reid, Miles. What is a flip? Scanned notes available at https:// homepages.warwick.ac.uk/^masda/3folds/what_flip.pdf, 1992.Google Scholar

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