Skeletons of stable maps II: Superabundant geometries

We implement new techniques involving Artin fans to study the realizability of tropical stable maps in superabundant combinatorial types. Our approach is to understand the skeleton of a fundamental object in logarithmic Gromov--Witten theory -- the stack of prestable maps to the Artin fan. This is used to examine the structure of the locus of realizable tropical curves and derive 3 principal consequences. First, we prove a realizability theorem for limits of families of tropical stable maps. Second, we extend the sufficiency of Speyer's well-spacedness condition to the case of curves with good reduction. Finally, we demonstrate the existence of liftable genus 1 superabundant tropical curves that violate the well-spacedness condition.Comment: 17 pages, 1 figure. v2 fixes a minor gap in the proof of Theorem D and adds details to the construction of the skeleton of a toroidal Artin stack. Minor clarifications throughout. To appear in Research in the Mathematical Science

Degenerations of toric varieties over valuation rings

We develop a theory of multi-stage degenerations of toric varieties over finite rank valuation rings, extending the Mumford--Gubler theory in rank one. Such degenerations are constructed from fan-like structures over totally ordered abelian groups of finite rank. Our main theorem describes the geometry of successive special fibers in the degeneration in terms of the polyhedral geometry of a system of recession complexes associated to the fan.Comment: 13 pages. v3: Added Example 4.1.8 and new references. To appear in Bulletin of the London Mathematical Societ

A note on cycles of curves in a product of pairs

We discuss the role of subdivisions of tropical moduli spaces in logarithmic Gromov-Witten theory, and use them to study the virtual class of curves in a product of pairs. Our main result is that the cycle-valued logarithmic Gromov-Witten theory of $X\times Y$ decomposes into a product of pieces coming from $X$ and $Y$, but this decomposition must be considered in a blowup of the moduli space of curves. This blowup is specified by tropical moduli data. As an application, we show that the cycle of curves in a toric variety with fixed contact orders is a product of virtual strict transforms of double ramification cycles. The formalism we outline offers a unified viewpoint on a number of recent results in logarithmic Gromov-Witten theory, including works of Herr, Holmes-Pixton-Schmitt, and Nabijou and the author.Comment: 12 pages. Comments are welcom