Tropical expansions and toric variety bundles
Francesca Carocci, Navid Nabijou
TL;DR
This work analyzes tropical expansions of a toroidal embedding $(X|D)$ and the relative geometry of the collapsing maps $\rho_v: Y_v \to X_v$. It proves that, over the interior $X_v^\circ$, each $\rho_v$ is a toric variety bundle and provides a fibrewise GIT construction to realize $Y_v^\circ$ from explicit line bundles on $X_v^\circ$, together with a criterion for extending this structure to all of $Y_v$. When the extension holds, the paper describes $Y_v$ as a fibrewise GIT quotient and, in general, offers a cut-and-paste description of $Y_v$ as a stratified union of toric bundles. The results are globalised via Artin fans and isotropic cone complexes, yielding a combinatorial recipe for constructing the toric bundle and a method to compute the mixing collection from piecewise-linear data. Overall, the work connects tropical data with explicit toric and GIT constructions, advancing the toolkit for tropical and logarithmic enumerative geometry and informing constructions in moduli spaces of expanded targets.
Abstract
A tropical expansion is a degeneration of a toroidal embedding, induced by a polyhedral subdivision of its tropicalisation. Each irreducible component of a tropical expansion admits a collapsing map down to a stratum of the original variety. We study the relative geometry of this map. We give a polyhedral criterion for the map to have the structure of a toric variety bundle, and prove that this structure always exists over the interior of the codomain. We give examples demonstrating that this is the strongest statement one can hope for in general. In addition, we provide a combinatorial recipe for constructing the toric variety bundle as a fibrewise GIT quotient of an explicit split vector bundle. Our proofs make systematic use of Artin fans as a language for globalising local toric models.