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Orbifold pseudo-effective cones of toric stacks

Ratko Darda, Takehiko Yasuda

TL;DR

The paper determines the orbifold pseudo-effective cone for split toric stacks by formulating the orbifold Néron–Severi space to include twisted sectors and establishing a duality with the orbifold moving cone via stacks of stacky curves. The authors construct explicit generators for the orbifold cone in terms of torus-invariant divisors and twisted-sector contributions, and they adapt Payne’s base-locus method to the stacky setting to realize all necessary classes. A detailed construction of orbifold numerical classes of stacky curves, their localization, and the action of the stacky torus on twisted arcs underpins the main result. The outcome provides a concrete, computable description of $\ar{Eff}_{\mathrm{orb}}(\\mathcal{X})$ for split toric stacks, with implications for extending Batyrev–Manin-type conjectures to orbifold contexts.

Abstract

In this paper, we explicitly describe the orbifold pseudo-effective cone of a split toric stack.

Orbifold pseudo-effective cones of toric stacks

TL;DR

The paper determines the orbifold pseudo-effective cone for split toric stacks by formulating the orbifold Néron–Severi space to include twisted sectors and establishing a duality with the orbifold moving cone via stacks of stacky curves. The authors construct explicit generators for the orbifold cone in terms of torus-invariant divisors and twisted-sector contributions, and they adapt Payne’s base-locus method to the stacky setting to realize all necessary classes. A detailed construction of orbifold numerical classes of stacky curves, their localization, and the action of the stacky torus on twisted arcs underpins the main result. The outcome provides a concrete, computable description of for split toric stacks, with implications for extending Batyrev–Manin-type conjectures to orbifold contexts.

Abstract

In this paper, we explicitly describe the orbifold pseudo-effective cone of a split toric stack.

Paper Structure

This paper contains 14 sections, 12 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Toric stacks and stacky fans
  3. Sectors and boxes
  4. Exact sequences
  5. "Non-orbifold" exact sequences
  6. "Orbifold" exact sequences
  7. Orbifold numerical classes of stacky curves
  8. $k\llparenthesis t\rrparenthesis$-points and twisted arcs
  9. The residue map
  10. Orbifold numerical classes
  11. Localization of orbifold numerical classes
  12. The orbifold class derived by a one-parameter subgroup
  13. The "action" of $\operatorname{\mathrm{J}}_{\infty}\mathcal{T}$ on $\mathcal{J}_{\infty}\mathcal{X}$
  14. Orbifold pseudo-effective cones

Key Result

Theorem 1.1

Let $\mathcal{E}_{\rho}\subset\mathcal{X}$ be the torus-invariant prime divisors on $\mathcal{X}$ corresponding to $\rho\in\Sigma(1)$ respectively and let $[\mathcal{E}_{\rho}]$ denote their numerical classes. Then, the orbifold pseudo-effective cone $\overline{\mathrm{Eff}}_{\mathrm{orb}}(\mathcal{ and

Theorems & Definitions (40)

  • Theorem 1.1: Corollary \ref{['cor:main']}
  • Example 1.2
  • Remark 2.1
  • Remark 4.1
  • Definition 5.1
  • Definition 5.2
  • Definition 5.3: darda2024thebatyrevtextendashmanin
  • Definition 5.4
  • Definition 5.5
  • Proposition 5.6
  • ...and 30 more