Fibration theorems for varieties with the weak Hilbert property

TL;DR

This paper introduces new fibration theorems for the weak Hilbert property (WHP) and its integral analogue, establishing conditions under which WHP is preserved in fibered varieties and applying these results to abelian schemes.

Contribution

It presents novel fibration theorems for WHP and integral WHP, extending previous results and providing tools to analyze the property in fibered varieties.

Findings

If the base has the strong Hilbert property and the fiber has WHP, then the total space also has WHP.

Certain abelian schemes over HP varieties possess WHP under the new criteria.

A new fibration theorem for integral WHP generalizes earlier product theorems.

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 $Z$.