Motivic Coh and Quot zeta functions of singular curves
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909893545426944 |
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| author | Huang, Yifeng Jiang, Ruofan |
| author_facet | Huang, Yifeng Jiang, Ruofan |
| contents | We present a general and effective algebraic framework for enumerating finite-length quotients of a torsion-free sheaf of arbitrary rank (the Quot zeta function) and finite-length coherent sheaves (the Coh zeta function) over reduced singular curves. We prove that Quot zeta functions are motivically rational, using a novel parametrization and the geometry of affine Grassmannians, and that they satisfy an arbitrary-rank reflection principle, via harmonic analysis. We show that the a normalized high-rank limit of Quot zeta functions converges to the Coh zeta function. As a first application, we compute explicit formulas for these zeta functions for all $y^2 = x^n$ singularities, revealing a surprising and previously unknown connection to Rogers--Ramanujan type $q$-series. Further applications to affine Springer fibers and commuting varieties are also discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_12528 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Motivic Coh and Quot zeta functions of singular curves Huang, Yifeng Jiang, Ruofan Algebraic Geometry Combinatorics Number Theory 14D20, 14M15, 11S45, 33D15 We present a general and effective algebraic framework for enumerating finite-length quotients of a torsion-free sheaf of arbitrary rank (the Quot zeta function) and finite-length coherent sheaves (the Coh zeta function) over reduced singular curves. We prove that Quot zeta functions are motivically rational, using a novel parametrization and the geometry of affine Grassmannians, and that they satisfy an arbitrary-rank reflection principle, via harmonic analysis. We show that the a normalized high-rank limit of Quot zeta functions converges to the Coh zeta function. As a first application, we compute explicit formulas for these zeta functions for all $y^2 = x^n$ singularities, revealing a surprising and previously unknown connection to Rogers--Ramanujan type $q$-series. Further applications to affine Springer fibers and commuting varieties are also discussed. |
| title | Motivic Coh and Quot zeta functions of singular curves |
| topic | Algebraic Geometry Combinatorics Number Theory 14D20, 14M15, 11S45, 33D15 |
| url | https://arxiv.org/abs/2312.12528 |