A HILBERT IRREDUCIBILITY THEOREM FOR ENRIQUES SURFACES

We define the over-exceptional lattice of a minimal algebraic surface of Kodaira dimension 0. Bounding the rank of this object, we prove that a conjecture by Campana [Ann. Inst. Fourier (Grenoble) 54 (2004), pp. 499-630] and Corvaja-Zannier [Math. Z. 286 (2017), pp. 579-602] holds for Enriques surfaces, as well as K3 surfaces of Picard rank >= 6 apart from a finite list of geometric Picard lattices. Concretely, we prove that such surfaces over finitely generated fields of characteristic 0 satisfy the weak Hilbert Property after a finite field extension of the base field. The degree of the field extension can be uniformly bounded.