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

Finn Bartsch

Abstract

We give an explicit example of a fibration $f \colon X \to Y$ between smooth projective varieties whose "orbifold base" $Δ_f$ in the sense of Campana has the property that the induced morphism $X \to (Y, Δ_f)$ 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 $(Y, Δ_f)$ 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.

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

Abstract

We give an explicit example of a fibration between smooth projective varieties whose "orbifold base" in the sense of Campana has the property that the induced morphism is not a morphism of C-pairs (i.e., it is not an "orbifold morphism"). We however also show that this cannot happen if is "neat" and 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.
Paper Structure (6 sections, 16 theorems, 2 equations)

This paper contains 6 sections, 16 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Examples
  3. Neat morphisms
  4. Applications
  5. Dense entire curves
  6. Density of integral points

Key Result

Theorem A

There exists a surjective morphism $f \colon X \to Y$ from a smooth projective threefold $X$ onto a smooth projective surface $Y$, with connected fibers, such that $X \to (Y, \Delta_f)$ is not a C-pair morphism.

Theorems & Definitions (39)

  • Theorem A: Section \ref{['sect:examples']}
  • Theorem B: Section \ref{['sect:neatness']}
  • Theorem C: Section \ref{['sect:applications']}
  • Theorem D: Section \ref{['sect:applications']}
  • Example 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • ...and 29 more