Classifying Toric and Semitoric Fans by Lifting Equations from SL₂(Z)
We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group SL₂(Z) to its preimage in the universal cover of SL₂(R) . With this method, we recover the classification of two-dimensional toric fans and obtain a description of their...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2018 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2018
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/209448 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Classifying Toric and Semitoric Fans by Lifting Equations from SL₂(Z) / D.M. Kane, J. Palmer, Á. Pelayo // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 48 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862684272650878976 |
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| author | Kane, D.M. Palmer, J. Pelayo, Á. |
| author_facet | Kane, D.M. Palmer, J. Pelayo, Á. |
| citation_txt | Classifying Toric and Semitoric Fans by Lifting Equations from SL₂(Z) / D.M. Kane, J. Palmer, Á. Pelayo // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 48 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group SL₂(Z) to its preimage in the universal cover of SL₂(R) . With this method, we recover the classification of two-dimensional toric fans and obtain a description of their semitoric analogue. As an application to symplectic geometry of Hamiltonian systems, we give a concise proof of the connectivity of the moduli space of toric integrable systems in dimension four, recovering a known result, and extend it to the case of semitoric integrable systems with a fixed number of focus-focus points and which are in the same twisting index class. In particular, we show that any semitoric system with precisely one focus-focus singular point can be continuously deformed into a system in the same isomorphism class as the Jaynes-Cummings model from optics.
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| first_indexed | 2025-12-07T15:57:37Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-209448 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T15:57:37Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kane, D.M. Palmer, J. Pelayo, Á. 2025-11-21T19:01:28Z 2018 Classifying Toric and Semitoric Fans by Lifting Equations from SL₂(Z) / D.M. Kane, J. Palmer, Á. Pelayo // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 48 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 52B20; 15B36; 53D05 arXiv: 1502.07698 https://nasplib.isofts.kiev.ua/handle/123456789/209448 https://doi.org/10.3842/SIGMA.2018.016 We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group SL₂(Z) to its preimage in the universal cover of SL₂(R) . With this method, we recover the classification of two-dimensional toric fans and obtain a description of their semitoric analogue. As an application to symplectic geometry of Hamiltonian systems, we give a concise proof of the connectivity of the moduli space of toric integrable systems in dimension four, recovering a known result, and extend it to the case of semitoric integrable systems with a fixed number of focus-focus points and which are in the same twisting index class. In particular, we show that any semitoric system with precisely one focus-focus singular point can be continuously deformed into a system in the same isomorphism class as the Jaynes-Cummings model from optics. We thank the anonymous referees for reading the paper carefully and providing very helpful suggestions, which have improved the paper. JP and ÁP were partially supported by NSF grants DMS-1055897 and DMS-1518420. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Classifying Toric and Semitoric Fans by Lifting Equations from SL₂(Z) Article published earlier |
| spellingShingle | Classifying Toric and Semitoric Fans by Lifting Equations from SL₂(Z) Kane, D.M. Palmer, J. Pelayo, Á. |
| title | Classifying Toric and Semitoric Fans by Lifting Equations from SL₂(Z) |
| title_full | Classifying Toric and Semitoric Fans by Lifting Equations from SL₂(Z) |
| title_fullStr | Classifying Toric and Semitoric Fans by Lifting Equations from SL₂(Z) |
| title_full_unstemmed | Classifying Toric and Semitoric Fans by Lifting Equations from SL₂(Z) |
| title_short | Classifying Toric and Semitoric Fans by Lifting Equations from SL₂(Z) |
| title_sort | classifying toric and semitoric fans by lifting equations from sl₂(z) |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209448 |
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