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An Effective Criterion for Covering Maps Between Real Varieties

Rizeng Chen

Abstract

In this paper, we show that a quasi-finite, flat morphism with locally constant geometric fibers between varieties over a real closed field induces a covering map on the rational points. This serves as an effective criteria for covering maps, as we show that these conditions can be checked by the algorithms developed in this paper.

An Effective Criterion for Covering Maps Between Real Varieties

Abstract

In this paper, we show that a quasi-finite, flat morphism with locally constant geometric fibers between varieties over a real closed field induces a covering map on the rational points. This serves as an effective criteria for covering maps, as we show that these conditions can be checked by the algorithms developed in this paper.
Paper Structure (19 sections, 18 theorems, 54 equations, 6 figures, 3 algorithms)

This paper contains 19 sections, 18 theorems, 54 equations, 6 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Previous Results
  4. Our Contribution
  5. Structure of the Paper
  6. Geometric Preliminaries
  7. Conventions
  8. Geometric Points and Geometric Fibers
  9. Flat and Étale Morphisms
  10. The Chow Form
  11. Subresultants
  12. Flat Morphism with Locally Constant Finite Geometric Fibers
  13. Semi-algebraic Covering Map
  14. Effective Aspect of the Criterion
  15. Covering Stratification, Sampling and Matrix Completion
  16. ...and 4 more sections

Key Result

Theorem \ref{thm-covering-map}

Let $\pi : X\to Y$ be a quasi-finite flat morphism with locally constant geometric fibers between $R$-varieties over a real closed field $R$. Then $\pi_R:X(R) \to Y(R)$ is a covering map in the Euclidean topology.

Figures (6)

  • Figure 1: Examples
  • Figure 2: Non-examples
  • Figure 3: $X$ as the intersection of zeros of defining equations
  • Figure 4: A finite étale double cover of the nodal curve
  • Figure 5: The number of real fibers in $D_+(u_0+u_1+u_2)$.
  • ...and 1 more figures

Theorems & Definitions (42)

  • Theorem \ref{thm-covering-map}
  • Example 1
  • Example 2
  • Theorem \ref{thm-reduced-g-etale-is-etale}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Proposition 3.1
  • proof
  • ...and 32 more