Existence of a flat projective fibration over A^1 with prescribed orbifold base and no divisible fibres

Ascertain whether there exists, for a given integer m > 1, a flat projective fibration V → A^1_\mathbb{C} whose orbifold base divisor is (1 − 1/m)[0] and which has no divisible fibres, providing a projective analogue of the affine flat fibration U → A^1_\mathbb{C} defined by (x_1,…,x_m) ↦ x_1^{m}⋯x_m^{2m−1}.

Background

The paper constructs an explicit affine flat fibration over A1_\mathbb{C} with orbifold divisor (1−1/m)[0] and no divisible fibres, which motivates the construction of Campana stacks and spaces.

The authors point out that extending this to a flat projective fibration with the same properties is unclear, noting that any projective compactification of the affine example would admit a section by Graber–Harris–Starr, which complicates such a construction.

References

It is unclear whether there exists a flat projective fibration $V \to \mathbb A1_{\mathbb C}$ with the same properties, as any projective extension $\overline U\to \mathbb A1_{\mathbb{C}}$ would admit a section by Graber--Harris--Starr .

— Weakly special varieties, Campana stacks, and Remarks on Orbifold Mordell  (2603.28745 - Bartsch et al., 30 Mar 2026) in Introduction, Campana stacks (paragraph after Theorem \ref{thm:U_to_X_and_orbifold_base})