Real semi-stable degenerations, real-oriented blow-ups and straightening corners

TL;DR

This paper investigates the topology of real parts of semi-stable degenerations, providing explicit descriptions and linking techniques to tropical geometry through real-oriented blow-ups.

Contribution

It introduces a general homeomorphism framework for real loci of degenerations using real-oriented blow-ups and applies it to toric degenerations and tropicalisation.

Findings

Homeomorphism description of real loci via real-oriented blow-up

Explicit stratified space descriptions of real fibres

Application to toric degenerations and tropicalisation

Abstract

We study totally real semi-stable degenerations (and more generally, smooth semi-stable degenerations). Our goal is to describe the homeomorphism type of the real locus $\mathbf{R} X_t$ of the general fibre in terms of the special fibre. We give a general homeomorphism statement via the real-oriented blow-up of the family. Using this, we give more explicit descriptions of $\mathbf{R} X_t$ as a stratified space glued from (covers of) strata of the special fibre. We also give relative versions of the statements and consider the example of toric degenerations in order to link the technique to tropicalisation.