ETD

• Geometria Diofantea

The Hilbert Property is a property generalizing the classical Hilbert Theorem, that every finite number of irreducible polynomials (on the rational numbers Q) admit a common specialization that maintains the irreducibility. The classical theorem of Hilbert is then equivalent to saying that rational varieties have the Hilbert Property. The main purpose of this thesis is to find new examples of non-rational varieties that have the Hilbert Property. The most interesting examples are some K3 surfaces with a double elliptic fibration, and some quotients of A^n by a finite group G (which are, usually, unirational varieties, but not rational), for which a positive answer to the Inverse Galois Problem is known.