Enumeration of Rational Cuspidal Curves via the WDVV equation
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909560960188416 |
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| author | Biswas, Indranil Choudhury, Apratim Mukherjee, Ritwik Paul, Anantadulal |
| author_facet | Biswas, Indranil Choudhury, Apratim Mukherjee, Ritwik Paul, Anantadulal |
| contents | We give a conjectural formula for the characteristic number of rational cuspidal curves in the projective plane by extending the idea of Kontsevich's recursion formula (namely, pulling back the equality of two divisors in the four pointed moduli space). The key geometric input that is needed here is that in the closure of rational cuspidal curves, there are two component rational curves which are tangent to each other at the nodal point. While this fact is geometrically quite believable, we haven't as yet proved it; hence our formula is for the moment conjectural. The answers that we obtain agree with what has been computed earlier Ran, Pandharipande, Zinger and Ernstrom and Kennedy. We extend this technique (modulo another conjecture) to obtain the characteristic number of rational quartics with an E6 singularity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10238 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Enumeration of Rational Cuspidal Curves via the WDVV equation Biswas, Indranil Choudhury, Apratim Mukherjee, Ritwik Paul, Anantadulal Algebraic Geometry Differential Geometry 14N35, 14J45 We give a conjectural formula for the characteristic number of rational cuspidal curves in the projective plane by extending the idea of Kontsevich's recursion formula (namely, pulling back the equality of two divisors in the four pointed moduli space). The key geometric input that is needed here is that in the closure of rational cuspidal curves, there are two component rational curves which are tangent to each other at the nodal point. While this fact is geometrically quite believable, we haven't as yet proved it; hence our formula is for the moment conjectural. The answers that we obtain agree with what has been computed earlier Ran, Pandharipande, Zinger and Ernstrom and Kennedy. We extend this technique (modulo another conjecture) to obtain the characteristic number of rational quartics with an E6 singularity. |
| title | Enumeration of Rational Cuspidal Curves via the WDVV equation |
| topic | Algebraic Geometry Differential Geometry 14N35, 14J45 |
| url | https://arxiv.org/abs/2409.10238 |