Nonspecial varieties and Generalized Lang-Vojta conjectures
Résumé
We construct a family of fibered threefolds $X_m \to (S,\Delta)$ that are weakly special but not special in the sense of Campana. We prove that if $m$ is big enough, the threefolds $X_m$ present behaviours that contradict the function field and analytic analogues of a conjecture of Abramovich and Colliot-Th\'el\`ene, thus providing evidences towards Campana's conjecture. This is obtained using an orbifold analogue of a result of Corvaja and Zannier in the function field and analytic setting, which we prove adapting the recent method of Ru and Vojta. We also formulate some generalizations of known conjectures on exceptional loci that fit into Campana's program and prove some cases over function fields.