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Hall Polynomials for Weighted projective lines
Representation Theory · Submitted 23 Feb 2025
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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)].
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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)].
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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 $f$. | background 0.92 | 0.14 | 0.04 | asserted 0.94 |
| s1 | We calculate Hall polynomials associated to extensions bundles, line bundles and torsion sheaves over $\mathbb{X}$. | contribution 0.70 | 0.96 | 0.35 | asserted 1.00 |
| s2 | 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)]. | interpretation 0.63 | 0.73 | 0.47 | asserted 1.00 |
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Original 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)].
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For example, the existence of Hall polynomials for representations of tame quivers was studied in [ref] , and Hall polynomials for some special classes of representations of tame quivers were calculated in [ref] .
Canonical bases for the quantum extended Kac-Moody algebras and Hall polynomials · Introduction · 0.94
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