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Geometric-combinatorial approaches to tilting theory for weighted projective lines

Jianmin Chen, Jinfeng Zhang

TL;DR

The paper presents a geometric-combinatorial framework for the category ${\rm coh}-\mathbb{X}$ of coherent sheaves on the weighted projective line of type $(2,2,n)$ by embedding it into a marked cylindrical surface with a order-$2$ symmetry. Indecomposable objects are in bijection with skew-curves on this surface, and maximal compatible collections of skew-arcs (pseudo-triangulations) correspond to tilting sheaves, with flips implementing tilting mutations. This yields a concrete, surface-based description of tilting objects and their mutations, and establishes the connectivity of the tilting graph ${\mathcal G}(\mathcal{T}_{\mathbb{X}})$. The results bridge representation theory with geometric combinatorics, and exploit an equivariant perspective linking ${\rm coh}-\mathbb{X}(2,2,n)$ to ${\rm coh}-\mathbb{X}(n,n)$ via a reflection symmetry, enabling explicit classifications and mutation dynamics. Overall, the work provides a robust toolkit for understanding tilting theory in tubular-weighted settings through a carefully constructed geometric model.

Abstract

We provide a geometric-combinatorial model for the category of coherent sheaves on the weighted projective line of type (2,2,n) via a cylindrical surface with n marked points on each of its upper and lower boundaries, equipped with an order 2 self-homeomorphism. A bijection is established between indecomposable sheaves on the weighted projective line and skew-curves on the surface. Moreover, by defining a skew-arc as a self-compatible skew-curve and a pseudo-triangulation as a maximal set of distinct pairwise compatible skew-arcs, we show that pseudo-triangulations correspond bijectively to tilting sheaves. Under this bijection, the flip of a skew-arc within a pseudo-triangulation coincides with the tilting mutation. As an application, we prove the connectivity of the tilting graph for the category of coherent sheaves.

Geometric-combinatorial approaches to tilting theory for weighted projective lines

TL;DR

The paper presents a geometric-combinatorial framework for the category of coherent sheaves on the weighted projective line of type by embedding it into a marked cylindrical surface with a order- symmetry. Indecomposable objects are in bijection with skew-curves on this surface, and maximal compatible collections of skew-arcs (pseudo-triangulations) correspond to tilting sheaves, with flips implementing tilting mutations. This yields a concrete, surface-based description of tilting objects and their mutations, and establishes the connectivity of the tilting graph . The results bridge representation theory with geometric combinatorics, and exploit an equivariant perspective linking to via a reflection symmetry, enabling explicit classifications and mutation dynamics. Overall, the work provides a robust toolkit for understanding tilting theory in tubular-weighted settings through a carefully constructed geometric model.

Abstract

We provide a geometric-combinatorial model for the category of coherent sheaves on the weighted projective line of type (2,2,n) via a cylindrical surface with n marked points on each of its upper and lower boundaries, equipped with an order 2 self-homeomorphism. A bijection is established between indecomposable sheaves on the weighted projective line and skew-curves on the surface. Moreover, by defining a skew-arc as a self-compatible skew-curve and a pseudo-triangulation as a maximal set of distinct pairwise compatible skew-arcs, we show that pseudo-triangulations correspond bijectively to tilting sheaves. Under this bijection, the flip of a skew-arc within a pseudo-triangulation coincides with the tilting mutation. As an application, we prove the connectivity of the tilting graph for the category of coherent sheaves.
Paper Structure (16 sections, 27 theorems, 77 equations, 20 figures, 3 tables)

This paper contains 16 sections, 27 theorems, 77 equations, 20 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Context
  3. Conventions
  4. Preliminaries
  5. Geometric model for the category ${\rm coh}\hbox{-}\mathbb{X}$
  6. A marked cylindrical surface with an order $2$ self-homeomorphism
  7. Skew-curves on $(\mathcal{S},M,\sigma)$
  8. The $\mathbb{L}(2,2,n)$-action on ${\rm vect}\hbox{-}\mathbb{X}$
  9. The geometric interpretation of tilting sheaf
  10. Tilting sheaves
  11. Compatibility of skew-curves
  12. Pseudo-triangulations and tilting bundles
  13. Flips and mutations
  14. Connectedness of the tilting graph $\mathcal{G}(\mathcal{T}_{\mathbb{X}})$
  15. The equivariant relationship between ${\rm coh}\hbox{-}\mathbb{X}$ and ${\rm coh}\hbox{-}\mathbb{Y}$
  16. ...and 1 more sections

Key Result

Theorem 1.1

(Proposition correspondence) The isoclasses in $\operatorname{ind}\left({\rm coh}\hbox{-}\mathbb{X}(2,2,n)\right)$ can be parameterized by the skew-curves in $\widehat{\mathbf{C}}$, through a bijection The explicit correspondence of $\widehat{\phi}$ is given in Table table:skew-curves-mapping.

Figures (20)

  • Figure 1: The cylinder $(\mathcal{S},M)$ and its universal cover $(\tilde{\mathcal{S}},\pi)$.
  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • ...and 15 more figures

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Proposition 3.5
  • proof
  • Example 3.6
  • Proposition 3.7
  • proof
  • Proposition 4.1
  • ...and 43 more