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

TL;DR

The paper constructs smooth projective threefolds fibred over $\mathbb{P}^1$ whose generic fibres are Enriques or K3 surfaces and which possess inf-multiple, non-divisible fibres. Using Lafon’s explicit Enriques surface, it shows these threefolds are weakly special but not special, and, for suitable base changes, yield orbifold bases of general type. Under the Orbifold Mordell Conjecture (which abc implies), these examples demonstrate that the Weakly Special Conjecture cannot hold, providing a counterexample to a long-standing conjecture. The work further proves non-isotriviality of Lafon’s Enriques surface, computes fundamental groups of the constructed spaces, and shows that certain K3 double covers extend over special fibres, revealing intricate degeneration phenomena and answering questions about divisibility and degenerations in Enriques and K3 fibrations. Collectively, the results connect arithmetic conjectures with geometric degeneration data and refute several complex-analytic analogues of the Weakly Special framework, while refining our understanding of orbifold bases and specialization behavior.

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 étale 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.

Theorems & Definitions (94)

• Definition 1.1
• Conjecture 1.2: Weakly Special Conjecture
• Theorem A
• Definition 1.3
• Theorem B: Corollary \ref{['corollary:first_theorem']}
• Theorem C: Consequence of Theorem \ref{['thm2:K3_version']}
• Theorem D: Consequence of Corollary \ref{['cor:enriques_fibration']}
• Conjecture 1.4: Orbifold Mordell
• Theorem 1.5: abc implies Orbifold Mordell
• Theorem E: Proved in Section \ref{['section:main_results']}