Ariyan Javan Peykar (Radboud University Nijmegen) : The weakly special conjecture contradicts abc

Lang conjectured that varieties of general type over a number field have very few rational points.  In 2000, guided by Lang's conjecture and in search of a converse statement, Abramovich, Colliot-Thelene, Harris, and Tschinkel formulated the "Weakly Special Conjecture": every weakly special variety over a number field has a potentially dense set of rational points. In this talk I will explain how this conjecture contradicts the abc conjecture, and more precisely Campana's "Orbifold Mordell" conjecture. Indeed, starting from an Enriques surface over Q(t) constructed by Lafon, we give the first examples of smooth projective weakly special threefolds which fiber over the projective line in Enriques surfaces   with nowhere reduced, but non-divisible, fibers. I will explain that the existence of these  threefolds shows that the Weakly Special Conjecture contradicts the abc conjecture. The existence of such threefolds also shows that Enriques surfaces and K3 surfaces can have non-divisible but nowhere reduced degenerations, thereby answering a question raised by Campana in 2005. This is joint work with Finn Bartsch, Frederic Campana, and Olivier Wittenberg.