Hall Polynomials for Weighted projective lines
Jiayi Chen, Bangming Deng, Shiquan Ruan
TL;DR
This work develops a unified method to compute Hall polynomials for coherent sheaves on weighted projective lines of type $(p_1,p_2,p_3)$ by leveraging orthogonal exceptional pairs and Green's formula. It treats line bundles, extension bundles $E_L\langle \vec{x}\rangle$, and torsion sheaves, deriving explicit polynomial expressions in the field size $q$ for various Hall numbers and polynomials, including recursive formulas. Through a derived equivalence with tame quivers, the results yield Hall polynomials for representations of tame quivers consistent with prior case-by-case analyses, providing a conceptual framework that unifies domestic, tubular, and wild types via the geometry of ${\mathbb X}$. The approach enhances the connectivity between Hall algebras, extension theory, and derived categories, with potential applications to canonical bases and quiver moduli. All formulas are expressed with explicit Euler-type invariants, enabling direct computation of Hall polynomials in these geometric settings.
Abstract
This paper deals with the triangle singularity defined by the \linebreak equation $f=X_1^{p_1}+X_2^{p_2}+X_3^{p_3}$ for weight triple $(p_1,p_2,p_3)$, as well as the category of coherent sheaves over the weighted projective line $\mathbb{X}$ defined by $f$. We calculate Hall polynomials associated to extensions bundles, line bundles and torsion sheaves over $\mathbb{X}$. By using derived equivalence, this provides a unified conceptual method for calculating Hall polynomials for representations of tame quivers obtained by Szántó and Szöllősi [J. Pure Appl. Alg. {\bf 228} (2024)].