Quadratic points on double planes

2025/11/03 by Chen, Nathan, Church, Ben, Pasten, Hector +1
#11G35 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2511.01669

Abstract

Zariski dense collections of quadratic points on curves X are well-understood by results of Harris--Silverman and Vojta, but when dim X ≥ 2 there is not an analogous geometric characterization, even conjecturally. In this note we consider the case of a double cover π\colon X → 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 → ℙ² with an additional source of dense quadratic points.

Citations

• Isolated and parameterized points on curves
• On the distribution of rational points on ramified covers of abelian varieties
• A more general abc conjecture

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