The Weakly Special Conjecture contradicts Orbifold Mordell, and hence the abc conjecture

Abstract

Starting from an Enriques surface over $\mathbb{Q}(t)$ constructed by Lafon, we give the first examples of smooth projective weakly special threefolds which fibre over the projective line in Enriques surfaces (resp. K3 surfaces) with nowhere reduced but non-divisible fibres and general type orbifold base. We verify that these families of Enriques surfaces (resp. K3 surfaces) are non-isotrivial and we compute their fundamental groups by studying the behaviour of local points along certain étale covers. The existence of these threefolds implies that the Weakly Special Conjecture formulated in 2000 contradicts the Orbifold Mordell Conjecture, and hence the abc conjecture. Using these examples, we can also easily disprove several complex-analytic analogues of the Weakly Special Conjecture. Finally, these examples show that Enriques surfaces and K3 surfaces can have non-divisible but nowhere reduced degenerations, thereby answering a question raised in 2005.