Hall Polynomials for Weighted projective lines

TL;DR

This paper computes Hall polynomials for coherent sheaves over weighted projective lines associated with triangle singularities, providing a unified approach linked to tame quiver representations.

Contribution

It introduces a method to calculate Hall polynomials for sheaves on weighted projective lines using derived equivalences, connecting to tame quiver representations.

Findings

Calculated Hall polynomials for various sheaves on weighted projective lines.

Unified method applicable to representations of tame quivers.

Links geometric categories to algebraic quiver representations.

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 Sz\'ant\'o and Sz\"oll\H{o}si [J. Pure Appl. Alg. {\bf 228} (2024)].