Zariski dense collections of quadratic points on curves \(X\) are well-understood by results of Harris–Silverman and Vojta, but when \(\dim X \ge 2\) there is not an analogous geometric characterization, even conjecturally. In this note we consider the case of a double cover \(\pi :X \rightarrow \mathbb {P}^r\) , 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 \(X \rightarrow \mathbb {P}^2\) with an additional source of dense quadratic points.
