Nonspecial varieties and generalised Lang–Vojta conjectures

Abstract

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.

Divisibility of polynomials and degeneracy of integral points

“…The goal of this project is to generalize the results of our previous paper [RTW21] to higher dimensions. In [RTW21] we dealt with two competing conjectures that aim to characterize algebraic varieties defined over a number field k that have a potentially dense set of k-rational points.…”

“…The goal of this project is to generalize the results of our previous paper [RTW21] to higher dimensions. In [RTW21] we dealt with two competing conjectures that aim to characterize algebraic varieties defined over a number field k that have a potentially dense set of k-rational points. On one hand Campana conjectured that the class of these varieties is the class of special varieties, introduced in [Cam04], while the Weak Specialness Conjecture (see [HT00, Conjecture 1.2]) predicts that these should be the weakly special varieties, i.e.…”

Hyperbolicity and specialness of symmetric powers

“…It has been shown in [CP07] and [RTW21] that one cannot replace "special" by "weakly-special" in Conjecture 3.7 in the analytic and function fields settings.…”

Lang–Vojta conjecture over function fields for surfaces dominating $${{\mathbb {G}}}_m^2$$

“…On the other hand, our theorem contains a constant C that does not appear in Vojta's formulation. For further discussion on the exceptional set we refer to [34,Section 3].…”