2013/12/10 by Krzysztof Jan Nowak, Nowak, Krzysztof Jan
Mathematics · #03C10 #14P10 #Advanced Topology and Set Theory #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Primary: 14G27 #Secondary: 12J25
paper · pdf · doi:10.48550/arxiv.1312.2935
openalex publication_date 2013/12/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper develops algebraic geometry over Henselian real valued (i.e. of rank 1) fields K, being a sequel to our paper about that over Henselian discretely valued fields. Several results are given including: a certain concept of fiber shrinking (a relaxed version of curve selection) for definable sets, the canonical projection Kn × Kℙm → Kn and blow-ups of the K-points of smooth K-varieties are definably closed maps, a descent property for blow-ups, a version of the Lojasiewicz inequality for continuous rational functions and the theorem on extending continuous hereditarily rational functions, established for the real and p-adic varieties in our joint paper with J. Kollar. The descent property enables application of desingularization and transformation to a normal crossing by blowing up in much the same way as over the locally compact ground field. Our approach applies quantifier elimination due to Pas.