Elliptic Fibrations and the Hilbert Property.

For a number field |$K$|⁠ , an algebraic variety |$X/K$| is said to have the Hilbert Property if |$X(K)$| is not thin. We are going to describe some examples of algebraic varieties, for which the Hilbert Property is a new result. The first class of examples is that of smooth cubic hypersurfaces with a |$K$| -rational point in |${\mathbb{P}}_n/K$|⁠ , for |$n \geq 3$|⁠. These fall in the class of unirational varieties, for which the Hilbert Property was conjectured by Colliot-Thélène and Sansuc. We then provide a sufficient condition for which a surface endowed with multiple elliptic fibrations has the Hilbert Property. As an application, we prove the Hilbert Property of a class of K3 surfaces, and some Kummer surfaces. [ABSTRACT FROM AUTHOR]