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Campana separable rational connectedness of toric orbifold

Enhao Feng, Sara Mehidi

TL;DR

This paper advances Campana's orbifold theory in the toric setting by proving separable Campana rational connectedness for smooth non-klt toric orbifolds and providing a detailed combinatorial framework for singular toric surfaces. It develops stable log maps and contact orders to translate geometric questions into lattice and determinant conditions, yielding explicit criteria and counterexamples in positive characteristic. In particular, it shows non-klt smooth toric orbifolds are separably CRC, gives a ray-based criterion for toric surfaces, and demonstrates that certain weighted projective spaces fail separability in characteristic $p$, while toric blow-ups can sometimes restore separability. These results deepen the understanding of Campana rational connectedness in singular and positive-characteristic toric contexts and have implications for arithmetic and birational geometry.

Abstract

We prove that smooth non-klt toric orbifolds are separably Campana rationally connected, extending the result in the klt case. We also show that there always exists a positive characteristic in which a singular weighted projective space, viewed as a non-klt Campana orbifold, is not separably Campana rationally connected.

Campana separable rational connectedness of toric orbifold

TL;DR

This paper advances Campana's orbifold theory in the toric setting by proving separable Campana rational connectedness for smooth non-klt toric orbifolds and providing a detailed combinatorial framework for singular toric surfaces. It develops stable log maps and contact orders to translate geometric questions into lattice and determinant conditions, yielding explicit criteria and counterexamples in positive characteristic. In particular, it shows non-klt smooth toric orbifolds are separably CRC, gives a ray-based criterion for toric surfaces, and demonstrates that certain weighted projective spaces fail separability in characteristic , while toric blow-ups can sometimes restore separability. These results deepen the understanding of Campana rational connectedness in singular and positive-characteristic toric contexts and have implications for arithmetic and birational geometry.

Abstract

We prove that smooth non-klt toric orbifolds are separably Campana rationally connected, extending the result in the klt case. We also show that there always exists a positive characteristic in which a singular weighted projective space, viewed as a non-klt Campana orbifold, is not separably Campana rationally connected.

Paper Structure

This paper contains 11 sections, 15 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Main results.
  3. Background
  4. Stable log map
  5. Contact order
  6. Campana Rational Connectedness for toric orbifolds.
  7. Smooth Campana non-klt toric orbifold
  8. Singular Campana non-klt toric orbifold
  9. Singular non-klt Campana toric surface
  10. Singularities of toric surfaces in characteristic $p$
  11. Weighted projective space

Key Result

Theorem 1.2

Let $(X, \Delta_{\epsilon})$ be a non-klt Campana toric orbifold over an algebraically closed field $\mathop{\mathrm{\textbf{k}}}\nolimits$ of characteristic $p > 0$. Suppose there exists a smooth cone in the fan defining $X$ such that all of its adjacent cones are smooth. Then $(X, \Delta_{\epsilon

Theorems & Definitions (40)

  • Conjecture 1.1: Campana
  • Theorem 1.2: Theorem \ref{['General smooth case']} and Corollary \ref{['3 adjacent smooth']}
  • Theorem 1.3: Theorem \ref{['Crit_sing']}
  • Theorem 1.4: Theorem \ref{['wps not sep']}
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3: Balancing condition
  • Example 2.4
  • Definition 2.5: Campana toric orbifold
  • Definition 2.6
  • ...and 30 more