A rational distance set is a subset of the plane such that the distance between any two points is a rational number. We show, assuming Lang's conjecture, that the cardinalities of rational distance sets in general position are uniformly bounded, generalizing results of Solymosi-de Zeeuw, Makhul-Shaffaf, Shaffaf and Tao. In the process, we give a criterion for certain varieties with non-canonical singularities to be of general type.

We prove that a complete intersection of c very general hypersurfaces of degrees d1, . . . , dc ≥ 2 in N -dimensional complex projective space is not ruled (and therefore not rational) provided that c i=1 di ≥ 2 3 N + c + 1. To this end we consider a degeneration to positive characteristic, following Kollár. Our argument does not require a resolution of the singularities of the special fiber of the degeneration. It relies on a generalization of Kollár's "algebraic Morse lemma" that controls the dimensions of the second-order Thom-Boardman singularities of general sections of Frobenius pullbacks of vector bundles.1 In the context of Theorem 0.3, it seems natural to define the singularity of a section s ∈ Γ(X, F * X E) at a point x ∈ X to be the equivalence class of e-tuples of power series consisting of all images of s under k-linear isomorphismsinduced by choices of formal coordinates around x and of OX,x-bases for E ⊗ OX,x.