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Fibration theorems for varieties with the weak Hilbert property

Sebastian Petersen

TL;DR

The paper establishes new fibration principles for the weak Hilbert property and its integral analogue for varieties over char$0$ fields. It proves that, for a smooth proper morphism $f:Y\to Z$, base $Z$ with $HP$ and generic fibre $Y_{R(Z)}$ with $WHP$ imply $Y$ has $WHP$, with a dual integral version when $f$ has a section and satisfies the extension property. The approach pivots on a key lemma describing vertical ramified covers and relies on a detailed analysis of pullbacks along sections and near-integral points, augmented by higher étale homotopy theory to relax properness assumptions. The results yield new instances of WHP for certain abelian schemes over HP bases and extend product-type and fibration-type theorems in the literature, offering tools for broader applications in arithmetic geometry. The work also discusses open problems and the influence of the base’s arithmetic geometry (e.g., Hermite–Minkowski schemes) on integrality phenomena.

Abstract

The weak Hilbert property (WHP) for varieties over fields of characteristic zero was introduced by Corvaja and Zannier in 2017. There exist integral variants of WHP for arithmetic schemes. We present new fibration theorems for both the WHP and its integral analogue. Our primary fibration result, in a sense dual to the mixed fibration theorems of Javanpeykar and Luger, establishes for a smooth proper morphism $f: Y \to Z$ of smooth connected varieties, that if $Z$ has the strong Hilbert property (HP) and the generic fiber has WHP, then the total space $Y$ also has WHP. As an application, we use this result in combination with previous work by Corvaja, Demeio, Javanpeykar, Lombardo, and Zannier and in combination with recent work of Javanpeykar to show that certain non-constant abelian schemes over HP varieties possess WHP. For integral WHP, we prove a new fibration theorem for proper smooth morphisms with a section, which generalizes earlier product theorems of Javanpeykar and Wittenberg, and of Luger. A key lemma gives information about the structure of covers of $Y$ whose branch locus is not dominant over $Z$.

Fibration theorems for varieties with the weak Hilbert property

TL;DR

The paper establishes new fibration principles for the weak Hilbert property and its integral analogue for varieties over char fields. It proves that, for a smooth proper morphism , base with and generic fibre with imply has , with a dual integral version when has a section and satisfies the extension property. The approach pivots on a key lemma describing vertical ramified covers and relies on a detailed analysis of pullbacks along sections and near-integral points, augmented by higher étale homotopy theory to relax properness assumptions. The results yield new instances of WHP for certain abelian schemes over HP bases and extend product-type and fibration-type theorems in the literature, offering tools for broader applications in arithmetic geometry. The work also discusses open problems and the influence of the base’s arithmetic geometry (e.g., Hermite–Minkowski schemes) on integrality phenomena.

Abstract

The weak Hilbert property (WHP) for varieties over fields of characteristic zero was introduced by Corvaja and Zannier in 2017. There exist integral variants of WHP for arithmetic schemes. We present new fibration theorems for both the WHP and its integral analogue. Our primary fibration result, in a sense dual to the mixed fibration theorems of Javanpeykar and Luger, establishes for a smooth proper morphism of smooth connected varieties, that if has the strong Hilbert property (HP) and the generic fiber has WHP, then the total space also has WHP. As an application, we use this result in combination with previous work by Corvaja, Demeio, Javanpeykar, Lombardo, and Zannier and in combination with recent work of Javanpeykar to show that certain non-constant abelian schemes over HP varieties possess WHP. For integral WHP, we prove a new fibration theorem for proper smooth morphisms with a section, which generalizes earlier product theorems of Javanpeykar and Wittenberg, and of Luger. A key lemma gives information about the structure of covers of whose branch locus is not dominant over .

Paper Structure

This paper contains 15 sections, 42 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Previous work
  3. Main results
  4. Strategy and outline of the paper
  5. Preliminaries
  6. Preliminaries on relative normalization
  7. Preliminaries on near integral points
  8. Classification of covers
  9. The extension property (EP)
  10. Splitting vertical covers
  11. Pullback of covers along sections
  12. Proof of Theorem \ref{['thm:1']}
  13. Proof of Theorem \ref{['thm:2']}
  14. Digression on étale homotopy theory
  15. Discussion of an open problem

Key Result

Theorem 1.6

(cf. Corollary coro:main:smoothproper) Let $K$ be a field of characteristic zero. Let $Y$ and $Z$ be connected smooth $K$-varieties. Let $f: Y\to Z$ be a smooth proper morphism. Assume that $Z$ has HP over $K$ and that the generic fibre $Y_{R(Z)}$ has WHP over the function field $R(Z)$ of $Z$. Then

Theorems & Definitions (110)

  • Definition 1.1
  • Conjecture 1.2
  • Conjecture 1.3
  • Definition 1.4
  • Conjecture 1.5
  • Theorem 1.6
  • Remark 1.7
  • Conjecture 1.8
  • Corollary 1.9
  • Example 1.10
  • ...and 100 more