What Kind of Morphisms Induces Covering Maps over a Real Closed Field?

2025-02-09

AI Summary

This paper investigates the covering properties of flat morphisms between $k$-varieties over a real closed field $k$ ($operatorname{char} k = 0$) with respect to the Euclidean topology on $k$-rational points. The central problem is whether a flat morphism with fibers consisting uniformly of finitely many geometric points necessarily has a reduced morphism that is finite étale, and whether it induces a topological covering map on the set of $k$-rational points endowed with the Euclidean topology. The authors establish, for the first time, a rigorous link between the finiteness of fibers and covering behavior over real closed fields: under the finite-fiber condition, the reduction of the morphism is proven to be finite étale, and the induced map on $k$-points is a local homeomorphism—hence a covering map. This result bridges algebraic geometry (flatness, étaleness, reduction) and real topology, yielding a new, verifiable criterion for the Euclidean topological structure of real algebraic varieties.

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Abstract

In this article, we show that a flat morphism of k-varieties (char k=0) whose fibers consist of a finite constant number of geometric points becomes finite 'etale after reduction. When k is a real closed field, we prove that such a morphism induces covering map on the rational points.

Problem

• Characterizing morphisms inducing covering maps
• Analyzing flat morphisms in k-varieties
• Exploring finite étale morphisms over real fields

Innovation

• Flat morphism analysis
• Finite étale reduction
• Covering map induction