Quadratic points on double planes

TL;DR

This paper explores the distribution of quadratic points on double covers of projective space, showing under Vojta's Conjecture that positivity of the canonical bundle limits dense quadratic points, with some exceptions.

Contribution

It connects Vojta's Conjecture to the scarcity of dense quadratic points on certain higher-dimensional varieties, extending understanding beyond curves.

Findings

Vojta's Conjecture implies finiteness of dense quadratic points for sufficiently positive canonical bundles.

Hilbert's Irreducibility Theorem ensures density of quadratic points in fibers of double covers.

Examples demonstrate additional sources of dense quadratic points on some surfaces.

Abstract

Zariski dense collections of quadratic points on curves $X$ are well-understood by results of Harris--Silverman and Vojta, but when $\dim X \geq 2$ there is not an analogous geometric characterization, even conjecturally. In this note we consider the case of a double cover $\pi \colon X \to \mathbb{P}^r$, where Hilbert's Irreducibility Theorem implies that the quadratic points in the fibers of $\pi$ are dense. We show that Vojta's Conjecture implies that, once the canonical bundle of $X$ is sufficiently positive, there are no other sources of Zariski dense quadratic points. This is complemented by several examples of surfaces $X \to \mathbb{P}^2$ with an additional source of dense quadratic points.