Tropical expansions and toric variety bundles

TL;DR

This paper investigates the structure of tropical expansions of toroidal embeddings, establishing conditions under which their irreducible components form toric variety bundles and providing a combinatorial construction method.

Contribution

It introduces a polyhedral criterion for when tropical expansion components are toric bundles and offers a GIT quotient construction for these bundles.

Findings

A polyhedral criterion for toric bundle structure.

Existence of toric bundle structure over the interior of the codomain.

A combinatorial recipe for constructing toric bundles as GIT quotients.

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.