Geometric-combinatorial approaches to tilting theory for weighted projective lines
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910782467342336 |
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| author | Chen, Jianmin Zhang, Jinfeng |
| author_facet | Chen, Jianmin Zhang, Jinfeng |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_06703 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric-combinatorial approaches to tilting theory for weighted projective lines Chen, Jianmin Zhang, Jinfeng Representation Theory Algebraic Geometry 14F06, 18E10, 05E10, 16S99, 57M50 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. |
| title | Geometric-combinatorial approaches to tilting theory for weighted projective lines |
| topic | Representation Theory Algebraic Geometry 14F06, 18E10, 05E10, 16S99, 57M50 |
| url | https://arxiv.org/abs/2501.06703 |