We construct a family of fibred threefolds $X_m \to (S , \Delta )$ such that $X_m$ has no étale cover that dominates a variety of general type but it dominates the orbifold $(S,\Delta )$ of general type. Following Campana, the threefolds $X_m$ are called weakly special but not special. The Weak Specialness Conjecture predicts that a weakly special variety defined over a number field has a potentially dense set of rational points. We prove that if m is big enough, the threefolds $X_m$ present behaviours that contradict the function field and analytic analogue of the Weak Specialness Conjecture. We prove our results by adapting the recent method of Ru and Vojta. We also formulate some generalisations of known conjectures on exceptional loci that fit into Campana’s program and prove some cases over function fields.

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 the nonsplit case of the Lang–Vojta conjecture over function fields for surfaces of log general type that are ramified covers of $${{\mathbb {G}}}_m^2$$ G m 2 . This extends the results of Corvaja and Zannier (J Differ Geom 93(3):355–377, 2013), where the conjecture was proved in the split case, and the results of Corvaja and Zannier (J Algebr Geom 17(2):295–333, 2008), Turchet (Trans Amer Math Soc 369(12):8537–8558, 2017) that were obtained in the case of the complement of a degree four and three component divisor in $${{\mathbb {P}}}^2$$ P 2 . We follow the strategy developed by Corvaja and Zannier and make explicit all the constants involved.

Abstract. Let f : (X, D) → B be a stable family with log canonical general fiber. We prove that, after a birational modification of the base B → B, there is a morphism from a high fibered power of the family to a pair of log general type. If in addition the general fiber is openly canonical, then there is a morphism from a high fibered power of the original family to a pair openly of log general type.

We prove the nonsplit case of the Lang-Vojta conjecture over function fields for surfaces of log general type that are ramified covers of G 2 m . This extends the results of [CZ13], where the conjecture was proved in the split case, and the results of [CZ08, Tur17] that were obtained in the case of the complement of a degree four and three component divisor in P 2 . We follow the strategy developed by Corvaja and Zannier and make explicit all the constants involved. Setting and Notations2.1. Function fields. In this paper we will denote by C a nonsingular projective curve (integral, separated scheme of finite type of dimension 1) defined over an algebraically closed field κ of characteristic zero and by S a finite set of points of C. We will denote by O S the ring of S-integers, i.e. the ring κ[C \ S] of regular functions in the complement of S: its elements are rational functions on C with poles contained in S. Similarly, we will denote by O * S the group of S-units, i.e. the group of invertible elements of O S : its elements are rational functions on C with both zeros and poles contained in S. If g(C) is the genus of the curve C, then the Euler characteristic of the affine curve C \ S, denoted by χ S (C), is defined as χ S (C) := χ(C \ S) = 2g(C) − 2 + #S.