Non-rational varieties with the Hilbert Property

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-Thélène 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.