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

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.

31 pages, 8 figures; added references and various corrections

References in corpus (4)

• Patchworking real algebraic varieties
• Euler Characteristic of real nondegenerate tropical complete intersections
• Euler characteristic and signature of real semi-stable degenerations
• Real phase structures on matroid fans and matroid orientations