A Hilbert Irreducibility Theorem for Enriques surfaces

TL;DR

This paper proves a version of the Hilbert irreducibility theorem for Enriques and certain K3 surfaces, showing they satisfy the weak Hilbert property after a bounded finite field extension.

Contribution

It introduces the over-exceptional lattice for minimal algebraic surfaces and proves the conjecture by Campana and Zannier for these surfaces, with uniform bounds.

Findings

Enriques surfaces satisfy the weak Hilbert property after finite extension.

Bound on the degree of the field extension is uniform.

The conjecture holds for K3 surfaces with Picard rank > 6.

Abstract

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 and Corvaja--Zannier holds for Enriques surfaces, as well as K3 surfaces of Picard rank greater than 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.