2023/12/19 by Yifeng Huang, Huang, Yifeng, Ruofan Jiang +1 · 1 citation
Mathematics · #11S45 #14D20 #14M15 #33D15 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2312.12528
openalex publication_date 2023/12/19 · openalex created_date 2023/12/22 · openalex updated_date 2026/07/28
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 y2 = xn 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.