The map to the orbifold base need not be an orbifold map

TL;DR

The paper provides an explicit example showing that the natural map to the orbifold base need not be an orbifold map, but this issue is avoided under certain conditions, impacting conjectures on dense entire curves and integral points.

Contribution

It constructs a specific example illustrating the failure of the orbifold map property and identifies conditions under which this failure cannot occur.

Findings

Explicit example of a non-orbifold map to the orbifold base.

Neat fibrations with well-behaved orbifold bases do not exhibit this failure.

Implications for Campana's conjectures on dense entire curves and integral points.

Abstract

We give an explicit example of a fibration $f \colon X \to Y$ between smooth projective varieties whose "orbifold base" $\Delta_f$ in the sense of Campana has the property that the induced morphism $X \to (Y, \Delta_f)$ is not a morphism of C-pairs (i.e., it is not an "orbifold morphism"). We however also show that this cannot happen if $f$ is "neat" and $(Y, \Delta_f)$ is sufficiently well-behaved. Finally, we discuss the implications of this statement towards conjectures of Campana aiming to give algebro-geometric characterizations of those varieties which either admit a dense entire curve or a potentially dense set of integral points.