Degenerations in tropical compactifications and tropical intersection theory of $\overline{M}_{0,n}$
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917415724515328 |
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| author | Griffin, Sean T. Levinson, Jake Ramadas, Rohini Silversmith, Rob |
| author_facet | Griffin, Sean T. Levinson, Jake Ramadas, Rohini Silversmith, Rob |
| contents | The main result of this paper is a formula for the limit cycle of a 1-parameter family of subvarieties of a tropical compactification, expressed in terms of tropical intersections. Our theorem generalizes results of Dickenstein-Feichtner-Sturmfels and Katz to the case of tropical compactifications.
In the second part of the paper, we apply our formula to the moduli space $\overline{M}_{0, n}$ of stable marked rational curves. We describe the tropicalization of the Kapranov maps $\overline{M}_{0, n}\to\mathbb{P}^{n-3}$, whose hyperplane pullbacks are the $ψ$-classes, with respect to a suitable choice of torus. We introduce tropical $ψ$-hypersurfaces (in genus zero). These are different from the standard definition of Mikhalkin and Kerber-Markwig, and may be of independent interest. We demonstrate our main result by giving a "firework algorithm" that computes limits of intersections of $ψ$-hypersurfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_15511 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Degenerations in tropical compactifications and tropical intersection theory of $\overline{M}_{0,n}$ Griffin, Sean T. Levinson, Jake Ramadas, Rohini Silversmith, Rob Algebraic Geometry 14T20, 14C17, 14H10, 14N10, 14M25 The main result of this paper is a formula for the limit cycle of a 1-parameter family of subvarieties of a tropical compactification, expressed in terms of tropical intersections. Our theorem generalizes results of Dickenstein-Feichtner-Sturmfels and Katz to the case of tropical compactifications. In the second part of the paper, we apply our formula to the moduli space $\overline{M}_{0, n}$ of stable marked rational curves. We describe the tropicalization of the Kapranov maps $\overline{M}_{0, n}\to\mathbb{P}^{n-3}$, whose hyperplane pullbacks are the $ψ$-classes, with respect to a suitable choice of torus. We introduce tropical $ψ$-hypersurfaces (in genus zero). These are different from the standard definition of Mikhalkin and Kerber-Markwig, and may be of independent interest. We demonstrate our main result by giving a "firework algorithm" that computes limits of intersections of $ψ$-hypersurfaces. |
| title | Degenerations in tropical compactifications and tropical intersection theory of $\overline{M}_{0,n}$ |
| topic | Algebraic Geometry 14T20, 14C17, 14H10, 14N10, 14M25 |
| url | https://arxiv.org/abs/2604.15511 |