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On two open questions for extension bundles

Qiang Dong, Shiquan Ruan

TL;DR

This paper resolves two open questions about extension bundles on weighted projective lines by (i) deriving an explicit orbit-counting framework for extension bundles under Picard and tau-actions using a Klein four-group, and (ii) constructing tilting objects in the stable category $\underline{\textup{vect}}-\mathbb{X}$ built from Auslander bundles for weight type $(2,p,q)$. The orbit analysis combines a canonical basis of the Grothendieck group, isomorphism criteria, and Burnside's lemma to yield exact counts for $|\mathcal{V}_2/\textup{Pic}(\mathbb{X})|$, $|\mathcal{V}_2/\mathbb{L}|$, and $|\mathcal{V}_2/\langle\tau\rangle|$, with transitivity results depending on weight types. The authors also establish invariants such as projective covers and injective hulls for extension bundles and provide explicit semistability criteria in terms of $\delta$-degrees, including tubular-type refinements. Finally, they construct tilting objects $T_1$ and $T_2$ from Auslander bundles for $(2,p,q)$, computing their endomorphism algebras and proving derived equivalences among related algebras, thereby linking extension-bundle structure to representation-theoretic and homological properties in the associated orbifold categories.

Abstract

In this paper we give positive answers for two open questions on extension bundles over weighted projective lines, raised by Kussin, Lenzing and Meltzer in the paper ``Triangle singularities, ADE-chains and weighted projective lines''.

On two open questions for extension bundles

TL;DR

This paper resolves two open questions about extension bundles on weighted projective lines by (i) deriving an explicit orbit-counting framework for extension bundles under Picard and tau-actions using a Klein four-group, and (ii) constructing tilting objects in the stable category built from Auslander bundles for weight type . The orbit analysis combines a canonical basis of the Grothendieck group, isomorphism criteria, and Burnside's lemma to yield exact counts for , , and , with transitivity results depending on weight types. The authors also establish invariants such as projective covers and injective hulls for extension bundles and provide explicit semistability criteria in terms of -degrees, including tubular-type refinements. Finally, they construct tilting objects and from Auslander bundles for , computing their endomorphism algebras and proving derived equivalences among related algebras, thereby linking extension-bundle structure to representation-theoretic and homological properties in the associated orbifold categories.

Abstract

In this paper we give positive answers for two open questions on extension bundles over weighted projective lines, raised by Kussin, Lenzing and Meltzer in the paper ``Triangle singularities, ADE-chains and weighted projective lines''.
Paper Structure (12 sections, 23 theorems, 74 equations)

This paper contains 12 sections, 23 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Orbit counting formula for extension bundles
  3. A canonical basis of the Grothendieck group
  4. Isomorphisms between extension bundles
  5. $\mathbb{L}$-orbit of extension bundles
  6. $\tau$-orbit of extension bundles
  7. Invariants and stabilities for extension bundles
  8. Projective cover and injective hull
  9. Semistable vector bundles
  10. Tilting objects in $\underline{\textup{vect}}\hbox{-}\mathbb{X}$ consisting of extension bundles
  11. A useful distinguished triangle
  12. Tilting objects in $\underline{\textup{vect}}\hbox{-}\mathbb{X}$

Key Result

Proposition 2.1

Assume that $\vec{x}=\sum\limits_{i=1}^{3}l_{i}\vec{x}_{i}+l\vec{c}$ is in normal form. Then we have

Theorems & Definitions (40)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • Theorem 2.5
  • proof
  • Corollary 2.6
  • ...and 30 more