Hilbert property for low-genus families and degree-one del Pezzo surfaces
Julian Demeio, Sam Streeter, Rosa Winter
TL;DR
The paper addresses the Hilbert property for unirational varieties by developing a framework that uses low-genus (genus 0 or 1) curve families, including non-linear ones, to spread rational points across a surface. It proves a general HP criterion (Theorem 1) for surfaces with multiple genus-1 fibrations and a geometrically integral low-genus family, provided the base is simply connected and the low-genus family has Zariski-dense rational points. Applying this to degree-one del Pezzo surfaces, it constructs a non-linear low-genus family on Nijgh's type surfaces and verifies HP under a non-torsion condition, using an elliptic fibration on a blown-up surface and a 3-to-1 map to a cubic surface. The authors further show that non-thin parameter sets exist and establish a universal family result (Theorem 3) giving HP on a non-thin subset of the parameter space with Picard rank 1 generically, thus obtaining HP for a broad class of minimal degree-one del Pezzo surfaces without conic fibrations. These results extend the known HP cases and link density techniques to a broader non-linear geometric framework.
Abstract
We prove that the Hilbert property is satisfied by certain del Pezzo surfaces of degree one and Picard rank 1 over fields finitely generated over $\mathbb{Q}$. We generalize results of the first author on elliptic surfaces and employ constructions used by Desjardins and the third author to prove density of rational points. Our results are the first on the Hilbert property for minimal del Pezzo surfaces of degree one without a conic fibration.