On varieties of Hilbert type

TL;DR

This paper proves that the property of being of Hilbert type is preserved under dominant morphisms with Hilbert type fibers and bases, extending known results to products and algebraic groups.

Contribution

It establishes a new criterion for varieties of Hilbert type based on dominant morphisms and applies it to products and algebraic groups.

Findings

Varieties of Hilbert type are preserved under certain dominant morphisms.

The result generalizes previous work on algebraic groups.

Answers a question of Serre regarding products of varieties.

Abstract

A variety X over a field K is of Hilbert type if the set of rational points X(K) is not thin. We prove that if f: X\to S is a dominant morphism of K-varieties and both S and all fibers f^{-1}(s), s in S(K), are of Hilbert type, then so is X. We apply this to answer a question of Serre on products of varieties and to generalize a result of Colliot-Th'el`ene and Sansuc on algebraic groups.