Tropical compactification and the Gromov--Witten theory of $\mathbb{P}^1$

Renzo Cavalieri; Hannah Markwig; Dhruv Ranganathan

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arXiv · 1410.2837

Tropical compactification and the Gromov--Witten theory of $\mathbb{P}^1$

Renzo Cavalieri·Hannah Markwig·Dhruv Ranganathan

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Abstract

We use tropical and nonarchimedean geometry to study the moduli space of genus $0$ stable maps to $\mathbb{P}^1$ relative to two points. This space is exhibited as a tropical compactification in a toric variety. Moreover, the fan of this toric variety may be interpreted as a moduli space for tropical relative stable maps with the same discrete data. As a consequence, we confirm an expectation of Bertram and the first two authors, that the tropical Hurwitz cycles are tropicalizations of classical Hurwitz cycles. As a second application, we obtain a full descendant correspondence for genus $0$ relative invariants of $\mathbb{P}^1$.

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Renzo Cavalieri, Hannah Markwig, Dhruv Ranganathan. 2014-10-10. Tropical compactification and the Gromov--Witten theory of $\mathbb{P}^1$. https://arxiv.org/abs/1410.2837

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