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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | https://arxiv.org/abs/2309.16112 |
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| _version_ | 1866916715441422336 |
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| author | Nakajima, Yusuke |
| author_facet | Nakajima, Yusuke |
| contents | A dimer model is a bipartite graph described on the real two-torus, and it gives the quiver as the dual graph. It is known that for any three-dimensional Gorenstein toric singularity, there exists a dimer model such that a GIT quotient parametrizing stable representations of the associated quiver is a projective crepant resolution of this singularity for some stability parameter. It is also known that the space of stability parameters has the wall-and-chamber structure, and for any projective crepant resolution of a three-dimensional Gorenstein toric singularity can be realized as the GIT quotient associated to a stability parameter contained in some chamber. In this paper, we consider dimer models giving rise to projective crepant resolutions of a toric compound Du Val singularity. We show that sequences of zigzag paths, which are special paths on a dimer model, determine the wall-and-chamber structure of the space of stability parameters. Moreover, we can track the variations of stable representations under wall-crossing using the sequences of zigzag paths. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_16112 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Variations of GIT quotients and dimer combinatorics for toric compound Du Val singularities Nakajima, Yusuke Algebraic Geometry Combinatorics Representation Theory A dimer model is a bipartite graph described on the real two-torus, and it gives the quiver as the dual graph. It is known that for any three-dimensional Gorenstein toric singularity, there exists a dimer model such that a GIT quotient parametrizing stable representations of the associated quiver is a projective crepant resolution of this singularity for some stability parameter. It is also known that the space of stability parameters has the wall-and-chamber structure, and for any projective crepant resolution of a three-dimensional Gorenstein toric singularity can be realized as the GIT quotient associated to a stability parameter contained in some chamber. In this paper, we consider dimer models giving rise to projective crepant resolutions of a toric compound Du Val singularity. We show that sequences of zigzag paths, which are special paths on a dimer model, determine the wall-and-chamber structure of the space of stability parameters. Moreover, we can track the variations of stable representations under wall-crossing using the sequences of zigzag paths. |
| title | Variations of GIT quotients and dimer combinatorics for toric compound Du Val singularities |
| topic | Algebraic Geometry Combinatorics Representation Theory |
| url | https://arxiv.org/abs/2309.16112 |