Skeletons of stable maps I: Rational curves in toric varieties

Abstract

We study the Berkovich analytification of the space of genus $0$ logarithmic stable maps to a toric variety $X$ and present applications to both algebraic and tropical geometry. On algebraic side, insights from tropical geometry give two new geometric descriptions of this space of maps -- (1) as an explicit toroidal modification of ${\overline{M}}_{0,n}\times X$ and (2) as a tropical compactification in a toric variety. On the combinatorial side, we prove that the tropicalization of the space of genus $0$ logarithmic stable maps coincides with the space of tropical stable maps, giving a large new collection of examples of faithful tropicalizations for moduli. Moreover, we identify the optimal settings in which the tropicalization of the moduli space of maps is faithful. The Nishinou--Siebert correspondence theorem is shown to be a consequence of this geometric connection between the algebraic and tropical moduli spaces.

28 pages, 5 TikZ figures. Final version to appear in Journal of the London Mathematical Society

References in corpus (5)

• Intersection Theory on Tropicalizations of Toroidal Embeddings
• Tropical geometry of moduli spaces of weighted stable curves
• Correspondence Theorems via Tropicalizations of Moduli Spaces
• Tropicalization of the moduli space of stable maps
• On the irreducibility of the space of genus zero stable log maps to wonderful compactifications

Cited by in corpus (9)

• A moduli stack of tropical curves
• Intersection Theory on Tropicalizations of Toroidal Embeddings
• The universal tropical Jacobian and the skeleton of the Esteves' universal Jacobian
• Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves
• Moduli of stable maps in genus one and logarithmic geometry II
• Enumerative geometry of elliptic curves on toric surfaces
• Tropical curves and covers and their moduli spaces
• Real Log Curves in Toric Varieties, Tropical Curves, and Log Welschinger Invariants
• Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds