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Hilbert properties of varieties

Arno Fehm, Ariyan Javanpeykar

TL;DR

This survey compiles the landscape of Hilbert-type properties for varieties, tracing from Hilbert's irreducibility theorem to modern notions of the Hilbert and weak Hilbert properties, including integral variants and potential density. It highlights preservation theorems under base change, morphisms, products, and fibrations, and surveys the HP/WHP status across curves, algebraic groups, surfaces, and key higher-dimensional classes. Central themes include the Noether program's geometric reinterpretation, Lang's conjectures, and Campana's special varieties as guiding principles for potential HP/WHP, with many precise results and numerous open problems. The work serves as a reference for which varieties are known to have HP/WHP or potential HP/WHP and where future breakthroughs are most needed.

Abstract

This is a survey of results on the Hilbert property of algebraic varieties, and variants of it.

Hilbert properties of varieties

TL;DR

This survey compiles the landscape of Hilbert-type properties for varieties, tracing from Hilbert's irreducibility theorem to modern notions of the Hilbert and weak Hilbert properties, including integral variants and potential density. It highlights preservation theorems under base change, morphisms, products, and fibrations, and surveys the HP/WHP status across curves, algebraic groups, surfaces, and key higher-dimensional classes. Central themes include the Noether program's geometric reinterpretation, Lang's conjectures, and Campana's special varieties as guiding principles for potential HP/WHP, with many precise results and numerous open problems. The work serves as a reference for which varieties are known to have HP/WHP or potential HP/WHP and where future breakthroughs are most needed.

Abstract

This is a survey of results on the Hilbert property of algebraic varieties, and variants of it.

Paper Structure

This paper contains 25 sections, 41 theorems, 1 equation.

Table of Contents

  1. Hilbert irreducibility
  2. Hilbert's theorem
  3. Hilbertian fields
  4. The Hilbert property
  5. The weak Hilbert property
  6. Integral Hilbert properties
  7. Potential density of rational points
  8. Preservation theorems
  9. Base change
  10. Going down morphisms
  11. Going up morphisms
  12. Product and fibration theorems
  13. Curves
  14. Algebraic groups
  15. Surfaces
  16. ...and 10 more sections

Key Result

Theorem 1.1

For every irreducible $f\in \mathbb{Q}[t,x]\setminus\mathbb{Q}[t]$ there exist infinitely many $\tau\in\mathbb{Q}$ with $f(\tau,x)$ irreducible in $\mathbb{Q}[x]$.

Theorems & Definitions (129)

  • Theorem 1.1
  • Remark 1.2
  • Example 1.3
  • Corollary 1.4
  • Example 1.5
  • Remark 1.6
  • Remark 1.7
  • Definition 1.8
  • Example 1.9
  • Theorem 1.10
  • ...and 119 more