Non-rational varieties with the Hilbert Property

A variety X over a field k is said to have the Hilbert Property if X(k) is not thin. We shall exhibit some examples of varieties, for which the Hilbert Property is a new result. We give a sufficient condition for descending the Hilbert Property to the quotient of a variety by the action of a finite group. Applying this result to linear actions of groups, we exhibit some examples of non-rational unirational varieties with the Hilbert Property, providing positive instances of a conjecture posed by Colliot-Theleene and Sansuc. We also give a sufficient condition for a surface with two elliptic fibrations to have the Hilbert Property, and use it to prove that a certain class of K3 surfaces have the Hilbert Property, generalizing a result of Corvaja and Zannier.