Density of rational points on del Pezzo surfaces of degree one

TL;DR

This paper establishes conditions under which the rational points on del Pezzo surfaces of degree one are dense, providing criteria that are verifiable and showing density results over real and rational fields.

Contribution

It introduces new geometric and arithmetic conditions ensuring Zariski density of rational points on degree one del Pezzo surfaces, including verifiable criteria and density results over Q.

Findings

Zariski density of rational points under specific geometric conditions

Density of such surfaces in the real topology within parameter spaces

Verifiable finite conditions for density on individual surfaces

Abstract

We state conditions under which the set S(k) of k-rational points on a del Pezzo surface S of degree 1 over an infinite field k of characteristic not equal to 2 or 3 is Zariski dense. For example, it suffices to require that the elliptic fibration over the projective line induced by the anticanonical map has a nodal fiber over a k-rational point. It also suffices to require the existence of a point in S(k) that does not lie on six exceptional curves of S and that has order 3 on its fiber of the elliptic fibration. This allows us to show that within a parameter space for del Pezzo surfaces of degree 1 over the field of real numbers, the set of surfaces S defined over the field Q of rational numbers for which the set S(Q) is Zariski dense, is dense with respect to the real analytic topology. We also include conditions that may be satisfied for every del Pezzo surface S and that can be verified with a finite computation for any del Pezzo surface S that does satisfy them.