Some results from algebraic geometry over Henselian real valued fields

Abstract

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 $K^n\times K{\mathbb{P}}^m\to K^n$ 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.

This paper was included in the article "Some results of algebraic geometry over Henselian rank one valued fields", published in Selecta Mathematica, DOI 10.1007/s00029-016-0245-y, arXiv:1410.3280 [math.AG]

References in corpus (1)

• Some results from algebraic geometry over complete discretely valued fields

Cited by in corpus (2)

• Some results of algebraic geometry over Henselian rank one valued fields
• Some results from algebraic geometry over complete discretely valued fields