Real Semi-Stable Degenerations, Real-Oriented Blow-Ups, and Straightening Corners

Abstract

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 ${\textbf R} X_t$ 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 ${\textbf R} X_t$ 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.