Classification of Rank 2 Cluster Varieties
We classify rank 2 cluster varieties (those for which the span of the rows of the exchange matrix is 2-dimensional) according to the deformation type of a generic fiber U of their X-spaces, as defined by Fock and Goncharov [Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 865-930]. Our approach is based on...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2019 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2019
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/210180 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Classification of Rank 2 Cluster Varieties / T. Mandel // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 21 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862665135568453632 |
|---|---|
| author | Mandel, T. |
| author_facet | Mandel, T. |
| citation_txt | Classification of Rank 2 Cluster Varieties / T. Mandel // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 21 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We classify rank 2 cluster varieties (those for which the span of the rows of the exchange matrix is 2-dimensional) according to the deformation type of a generic fiber U of their X-spaces, as defined by Fock and Goncharov [Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 865-930]. Our approach is based on the work of Gross, Hacking, and Keel for cluster varieties and log Calabi-Yau surfaces. Call U positive if dim[Γ(U, OU)]=dim(U) (which equals 2 in these rank 2 cases). This is the condition for the Gross-Hacking-Keel construction [Publ. Math. Inst. Hautes Études Sci. 122 (2015), 65-168] to produce an additive basis of theta functions on Γ(U, OU). We find that U is positive and either finite-type or non-acyclic (in the usual cluster sense) if and only if the inverse monodromy of the tropicalization Uᵗʳᵒᵖ of U is one of Kodaira's monodromies. In these cases, we prove uniqueness results about the log Calabi-Yau surfaces whose tropicalization is Uᵗʳᵒᵖ. We also describe the action of the cluster modular group on Uᵗʳᵒᵖ in the positive cases.
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| first_indexed | 2025-12-07T21:24:39Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-210180 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:24:39Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Mandel, T. 2025-12-03T14:26:56Z 2019 Classification of Rank 2 Cluster Varieties / T. Mandel // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 21 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 13F60; 14J32 arXiv: 1407.6241 https://nasplib.isofts.kiev.ua/handle/123456789/210180 https://doi.org/10.3842/SIGMA.2019.042 We classify rank 2 cluster varieties (those for which the span of the rows of the exchange matrix is 2-dimensional) according to the deformation type of a generic fiber U of their X-spaces, as defined by Fock and Goncharov [Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 865-930]. Our approach is based on the work of Gross, Hacking, and Keel for cluster varieties and log Calabi-Yau surfaces. Call U positive if dim[Γ(U, OU)]=dim(U) (which equals 2 in these rank 2 cases). This is the condition for the Gross-Hacking-Keel construction [Publ. Math. Inst. Hautes Études Sci. 122 (2015), 65-168] to produce an additive basis of theta functions on Γ(U, OU). We find that U is positive and either finite-type or non-acyclic (in the usual cluster sense) if and only if the inverse monodromy of the tropicalization Uᵗʳᵒᵖ of U is one of Kodaira's monodromies. In these cases, we prove uniqueness results about the log Calabi-Yau surfaces whose tropicalization is Uᵗʳᵒᵖ. We also describe the action of the cluster modular group on Uᵗʳᵒᵖ in the positive cases. Most of this paper is based on part of the author’s thesis, which was written while in graduate school at the University of Texas at Austin. I would like to thank my advisor, Sean Keel, for introducing me to these topics and for all his suggestions, insights, and support. I would also like to thank Yan Zhou for asking insightful questions that helped improve the final draft, and also the anonymous referees for their many very helpful suggestions. This work was supported in part by the Center of Excellence grant "Centre for Quantum Geometry of Moduli Spaces" from the Danish National Research Foundation (DNRF95), and later by the National Science Foundation RTG Grant DMS-1246989, and finally by the Starter Grant "Categorified Donaldson-Thomas Theory" no. 759967 of the European Research Council. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Classification of Rank 2 Cluster Varieties Article published earlier |
| spellingShingle | Classification of Rank 2 Cluster Varieties Mandel, T. |
| title | Classification of Rank 2 Cluster Varieties |
| title_full | Classification of Rank 2 Cluster Varieties |
| title_fullStr | Classification of Rank 2 Cluster Varieties |
| title_full_unstemmed | Classification of Rank 2 Cluster Varieties |
| title_short | Classification of Rank 2 Cluster Varieties |
| title_sort | classification of rank 2 cluster varieties |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210180 |
| work_keys_str_mv | AT mandelt classificationofrank2clustervarieties |