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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2508.14206 |
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| _version_ | 1866916909343047680 |
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| author | Ebrahimian, Parisa |
| author_facet | Ebrahimian, Parisa |
| contents | Tyomkin's correspondence theorem states the equality of counts of rational curves of fixed homology class in a toric surface satisfying point and cross-ratio conditions with their tropical counterparts. Such correspondence theorems allow us to derive non-tropical results from tropical ones; for example, Mikhalkin's correspondence theorem is used in the tropical proof of the famous Kontsevich formula for counts of plane rational curves of degree $d$ satisfying point conditions. This formula has been generalized to counts of curves in the Hirzebruch surface $\mathbb{F}_{2}$ satisfying point conditions. Further generalizations allow curves in $\mathbb{P}^2$ to satisfy multiple cross-ratio conditions. In this paper, we present a Kontsevich-style formula for the Hirzebruch surface $\mathbb{F}_r$, $r \in \mathbb{N}$, which counts rational tropical curves of a fixed homology class satisfying point and multiple cross-ratio conditions using tropical methods. Moreover, the cross-ratio conditions we impose on the curves allow more freedom. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_14206 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | General Kontsevich-style formula for Hirzebruch Surfaces Ebrahimian, Parisa Algebraic Geometry Combinatorics 14T15, 14T20, 14N10, 14N35 Tyomkin's correspondence theorem states the equality of counts of rational curves of fixed homology class in a toric surface satisfying point and cross-ratio conditions with their tropical counterparts. Such correspondence theorems allow us to derive non-tropical results from tropical ones; for example, Mikhalkin's correspondence theorem is used in the tropical proof of the famous Kontsevich formula for counts of plane rational curves of degree $d$ satisfying point conditions. This formula has been generalized to counts of curves in the Hirzebruch surface $\mathbb{F}_{2}$ satisfying point conditions. Further generalizations allow curves in $\mathbb{P}^2$ to satisfy multiple cross-ratio conditions. In this paper, we present a Kontsevich-style formula for the Hirzebruch surface $\mathbb{F}_r$, $r \in \mathbb{N}$, which counts rational tropical curves of a fixed homology class satisfying point and multiple cross-ratio conditions using tropical methods. Moreover, the cross-ratio conditions we impose on the curves allow more freedom. |
| title | General Kontsevich-style formula for Hirzebruch Surfaces |
| topic | Algebraic Geometry Combinatorics 14T15, 14T20, 14N10, 14N35 |
| url | https://arxiv.org/abs/2508.14206 |