Tropical compactification and the Gromov--Witten theory of $\mathbb{P}^1$
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 $\mathbb{P}^1$.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2014
- DOI:
- 10.48550/arXiv.1410.2837
- arXiv:
- arXiv:1410.2837
- Bibcode:
- 2014arXiv1410.2837C
- Keywords:
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- Mathematics - Algebraic Geometry;
- Mathematics - Combinatorics
- E-Print:
- 29 pages, 7 TikZ figures. v2: Minor revisions. To appear in Selecta Mathematica