Abstract
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.
Highlights
A fundamental problem in Diophantine geometry is to describe the distribution of rational points in an algebraic variety defined over a number field
The Green–Griffiths–Lang conjecture predicts that entire curves in varieties of general type should be contained in a proper Zariski closed subset, the already mentioned exceptional set
Let be a projective variety defined over a field
Summary
A fundamental problem in Diophantine geometry is to describe the distribution of rational points in an algebraic variety defined over a number field. The Green–Griffiths–Lang conjecture predicts that entire curves in varieties of general type should be contained in a proper Zariski closed subset, the already mentioned exceptional set Following this analogy, Campana has conjectured that specialness (and potential density) should correspond to the existence of Zariski dense entire curves (see [Cam04]). The pairs ( , Δ ) can be seen as orbifold generalisations of surfaces appearing in [CZ10], which provides examples of connected quasi-projective surfaces with a non-Zariski dense set of integral points For this class of examples we show that the function-field and analytic Weak Specialness Conjectures fail.
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