Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hilbert properties under base change in small extensions

Lior Bary-Soroker, Arno Fehm, Sebastian Petersen

TL;DR

This paper investigates how the Hilbert property (HP) and the weak Hilbert property (WHP) behave under base change along field extensions. It develops a thin/strongly thin framework and spreading-out techniques to prove preservation results for finitely generated and small extensions, and it derives a product-type perspective via a fibration approach. The authors establish that HP and WHP are preserved under these base-change scenarios, but also provide counterexamples showing limitations: HP/WHP do not always descend along arbitrary finite extensions and may fail to behave well under unions of chains. The results clarify the scope of base-change phenomena for HP and WHP in characteristic zero and highlight the nuanced differences between HP and WHP in various extension settings.

Abstract

We study the preservation of the Hilbert property and of the weak Hilbert property under base change in field extensions. In particular we show that these properties are preserved if the extension is finitely generated or Galois with finitely generated Galois group, and we also obtain some negative results.

Hilbert properties under base change in small extensions

TL;DR

This paper investigates how the Hilbert property (HP) and the weak Hilbert property (WHP) behave under base change along field extensions. It develops a thin/strongly thin framework and spreading-out techniques to prove preservation results for finitely generated and small extensions, and it derives a product-type perspective via a fibration approach. The authors establish that HP and WHP are preserved under these base-change scenarios, but also provide counterexamples showing limitations: HP/WHP do not always descend along arbitrary finite extensions and may fail to behave well under unions of chains. The results clarify the scope of base-change phenomena for HP and WHP in characteristic zero and highlight the nuanced differences between HP and WHP in various extension settings.

Abstract

We study the preservation of the Hilbert property and of the weak Hilbert property under base change in field extensions. In particular we show that these properties are preserved if the extension is finitely generated or Galois with finitely generated Galois group, and we also obtain some negative results.
Paper Structure (5 sections, 18 theorems, 6 equations)

This paper contains 5 sections, 18 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Finite extensions
  3. Finitely generated extensions
  4. Small extensions
  5. Counterexamples

Key Result

Theorem 1.1

Let $K$ be a field of characteristic zero, $X$ a normal $K$-variety, and $L/K$ a finitely generated field extension. If $X$ has HP (respectively WHP), then so does $X_L$.

Theorems & Definitions (34)

  • Theorem 1.1: cf. Cor. \ref{['cor:fg']}
  • Theorem 1.2: cf. Cor. \ref{['cor:small']}
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • Proposition 3.4
  • ...and 24 more