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Constructing Fano 3-folds from cluster varieties of rank 2

Part of: Surfaces and higher-dimensional varieties Arithmetic rings and other special rings

Published online by Cambridge University Press:  09 November 2020

Stephen Coughlan  and
Tom Ducat
Stephen Coughlan
Affiliation:
Lehrstuhl Mathematik VIII, Mathematisches Institut, Universitätsstrasse 30, 95447Bayreuth, Germanystephen.coughlan@uni-bayreuth.de
Tom Ducat
Affiliation:
School of Mathematics, University of Bristol, BristolBS8 1TW, UKt.ducat19@imperial.ac.uk The Heilbronn Institute for Mathematical Research, Bristol, UK
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Abstract

Cluster algebras give rise to a class of Gorenstein rings which enjoy a large amount of symmetry. Concentrating on the rank 2 cases, we show how cluster varieties can be used to construct many interesting projective algebraic varieties. Our main application is then to construct hundreds of families of Fano 3-folds in codimensions 4 and 5. In particular, for Fano 3-folds in codimension 4 we construct at least one family for 187 of the 206 possible Hilbert polynomials contained in the Graded Ring Database.

Keywords

Fano 3-foldscluster varietieskey varieties

MSC classification

Primary: 13F60: Cluster algebras
Secondary: 14J45: Fano varieties
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 9 , September 2020 , pp. 1873 - 1914
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Copyright
Copyright © The Author(s) 2020

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Footnotes

Current address: Department of Mathematics, Imperial College, South Kensington Campus, Huxley Building, 180 Queen's Gate, London, SW7 2AZ, UK

To Miles Reid on his 70th birthday.

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