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 ${\mathbb{P}}^1$.

arXiv comment29 pages, 7 TikZ figures. v2: Minor revisions. To appear in Selecta Mathematica