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 P-n/K, for n >= 3. These fall in the class of unirational varieties, for which the Hilbert Property was conjectured by Colliot-Thelene 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.

Automated summary

This article discusses the application of the Hilbert Property in mathematics, exploring examples of algebraic varieties and the relationship between varieties and the Hilbert Property.