Counting Curves with Tangencies
Interpreting tangency as a limit of two transverse intersections, we obtain a concrete formula to enumerate smooth degree plane curves tangent to a given line at multiple points with arbitrary order of tangency. Extending that idea, we then enumerate curves with one node with multiple tangencies to...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2025 |
| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2025
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/212879 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Counting Curves with Tangencies. Indranil Biswas, Apratim Choudhury, Ritwik Mukherjee and Anantadulal Paul. SIGMA 21 (2025), 012, 50 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862549707696373760 |
|---|---|
| author | Biswas, Indranil Choudhury, Apratim Mukherjee, Ritwik Paul, Anantadulal |
| author_facet | Biswas, Indranil Choudhury, Apratim Mukherjee, Ritwik Paul, Anantadulal |
| citation_txt | Counting Curves with Tangencies. Indranil Biswas, Apratim Choudhury, Ritwik Mukherjee and Anantadulal Paul. SIGMA 21 (2025), 012, 50 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Interpreting tangency as a limit of two transverse intersections, we obtain a concrete formula to enumerate smooth degree plane curves tangent to a given line at multiple points with arbitrary order of tangency. Extending that idea, we then enumerate curves with one node with multiple tangencies to a given line of any order. Subsequently, we enumerate curves with one cusp that are tangent to first order to a given line at multiple points. We also present a new way to enumerate curves with one node; it is interpreted as a degeneration of a curve tangent to a given line. That method is extended to enumerate curves with two nodes, and also curves with one tacnode are enumerated. In the final part of the paper, it is shown how this idea can be applied in the setting of stable maps and perform a concrete computation to enumerate rational curves with first-order tangency. A large number of low-degree cases have been worked out explicitly.
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| first_indexed | 2026-03-21T11:38:30Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-212879 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T11:38:30Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Biswas, Indranil Choudhury, Apratim Mukherjee, Ritwik Paul, Anantadulal 2026-02-13T13:49:49Z 2025 Counting Curves with Tangencies. Indranil Biswas, Apratim Choudhury, Ritwik Mukherjee and Anantadulal Paul. SIGMA 21 (2025), 012, 50 pages 1815-0659 2020 Mathematics Subject Classification: 14N35; 14J45; 53D45 arXiv:2312.10759 https://nasplib.isofts.kiev.ua/handle/123456789/212879 https://doi.org/10.3842/SIGMA.2025.012 Interpreting tangency as a limit of two transverse intersections, we obtain a concrete formula to enumerate smooth degree plane curves tangent to a given line at multiple points with arbitrary order of tangency. Extending that idea, we then enumerate curves with one node with multiple tangencies to a given line of any order. Subsequently, we enumerate curves with one cusp that are tangent to first order to a given line at multiple points. We also present a new way to enumerate curves with one node; it is interpreted as a degeneration of a curve tangent to a given line. That method is extended to enumerate curves with two nodes, and also curves with one tacnode are enumerated. In the final part of the paper, it is shown how this idea can be applied in the setting of stable maps and perform a concrete computation to enumerate rational curves with first-order tangency. A large number of low-degree cases have been worked out explicitly. We are very grateful to the referees for giving us constructive and detailed comments on the earlier version of the manuscript. We are grateful to Chitrabhanu Chaudhuri for several useful discussions related to this paper. We also thank Soumya Pal for writing a Python program to implement the Caporaso–Harris formula for verification. The first author is partially supported by a J.C. Bose Fellowship (JBR/2023/000003). The second author is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy– The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689). The fourth author would like to acknowledge the support of the Department of Atomic Energy, Government of India, under project no. RTI4001. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Counting Curves with Tangencies Article published earlier |
| spellingShingle | Counting Curves with Tangencies Biswas, Indranil Choudhury, Apratim Mukherjee, Ritwik Paul, Anantadulal |
| title | Counting Curves with Tangencies |
| title_full | Counting Curves with Tangencies |
| title_fullStr | Counting Curves with Tangencies |
| title_full_unstemmed | Counting Curves with Tangencies |
| title_short | Counting Curves with Tangencies |
| title_sort | counting curves with tangencies |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212879 |
| work_keys_str_mv | AT biswasindranil countingcurveswithtangencies AT choudhuryapratim countingcurveswithtangencies AT mukherjeeritwik countingcurveswithtangencies AT paulanantadulal countingcurveswithtangencies |