Degenerations of toric varieties over valuation rings

TL;DR

This paper extends the theory of degenerations of toric varieties over valuation rings to higher ranks, describing their geometry through polyhedral structures and recession complexes.

Contribution

It introduces a multi-stage degeneration framework for toric varieties over finite rank valuation rings, generalizing rank one theories with new geometric descriptions.

Findings

Describes the geometry of successive special fibers in degenerations.

Develops fan-like structures over totally ordered abelian groups.

Connects degeneration geometry with polyhedral recession complexes.

Abstract

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.