Quadratic points on double planes

TL;DR

This work studies the Zariski density of quadratic points on higher-dimensional varieties X that arise as double (cyclic) covers of projective space. Assuming Vojta's conjecture, and when the branch divisor has degree $2m$ with $m \ge r+4$ (so the canonical bundle becomes sufficiently positive), the authors show that every degree-$d$ point not contracted by the cover map π: X → $\mathbb{P}^r$ cannot be Zariski dense; in particular, for $d=2$ and $e=2$ this recovers the threshold $m \ge r+4$. The paper also constructs explicit examples of surfaces $X \to \mathbb{P}^2$ with an additional source of dense quadratic points, illustrating the sharpness of the bounds in certain cases, and discusses consequences and potential obstructions under Bombieri–Lang for ruling out secondary sources. Overall, the results connect Vojta-type height–discriminant inequalities to the geometry of branched covers and to concrete density phenomena for quadratic points in higher dimensions.

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 $π\colon X \to \mathbb{P}^r$, where Hilbert's Irreducibility Theorem implies that the quadratic points in the fibers of $π$ 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.

Theorems & Definitions (17)

• Theorem 1
• Corollary 2
• Theorem 3
• Lemma 2.1: Silverman
• proof
• Conjecture 2.2: vojta_diophantine, vojta_abc
• proof : Proof of Corollary \ref{['cor:small_gcd']}
• Example 4.1: $m=1$
• Example 4.2: $m=3$
• Lemma 4.3