Non-rational varieties with the Hilbert Property

TL;DR

This paper explores the Hilbert Property in algebraic varieties, providing new examples, criteria for quotients, and focusing on K3 surfaces with elliptic fibrations, contributing to understanding rational points in algebraic geometry.

Contribution

It introduces a criterion linking the Hilbert Property of a variety to its quotients and applies this to non-rational unirational varieties and K3 surfaces.

Findings

Examples of varieties with the Hilbert Property are described.

A criterion for the Hilbert Property of quotients is established.

Explicit examples of K3 surfaces with the Hilbert Property are provided.

Abstract

A variety $X/k$ is said to have the Hilbert Property if $X(k)$ is not thin. We shall describe some examples of varieties, for which the Hilbert Property is a new result. We give a criterion for determining when the Hilbert Property for a variety $X$ implies the Hilbert Property for quotients $X/G$ of the variety by an action of a finite group. In the case of linear actions of the group $G$, this gives examples of (non-rational) unirational varieties with the Hilbert Property, providing positive examples to a conjecture by Colliot-Th\'el\`ene and Sansuc. We focus then on the study of the Hilbert Property for K3 surfaces that have two elliptic fibrations, in particular on diagonal quartic surfaces, i.e. varieties of the form $ax^4+by^4+cz^4+dw^4=0$. We then show, through an explicit application, how one may use the criterion above to provide other examples of K3 surfaces with the Hilbert Property. Since the Hilbert Property is related to an abudance of rational points, K3 surfaces should (conjecturally) represent a limiting case in dimension 2.