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

TL;DR

This paper constructs specific threefolds fibered over the projective line with Enriques and K3 surfaces, demonstrating that the Weakly Special Conjecture conflicts with the Orbifold Mordell and abc conjectures, and addressing questions about degenerations.

Contribution

It provides the first examples of such threefolds with particular fiber properties, showing the inconsistency of the Weakly Special Conjecture with fundamental conjectures in number theory.

Findings

Weakly special threefolds with non-divisible, nowhere reduced fibers constructed

These examples contradict the Orbifold Mordell Conjecture and the abc conjecture

Enriques and K3 surfaces can have non-divisible, nowhere reduced degenerations

Abstract

Starting from an Enriques surface over $\mathbb{Q}(t)$ considered 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 compute their fundamental groups by studying the behaviour of local points along certain \'etale covers. The existence of the above 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, the existence of such threefolds shows that Enriques surfaces and K3 surfaces can have non-divisible but nowhere reduced degenerations, thereby answering a question raised in 2005.