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arXiv · 2102.04584

The braid group action on exceptional sequences for weighted projective lines

Edson R. Alvares·Eduardo N. Marcos·Hagen Meltzer
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Abstract

We give a new and intrinsic proof of the transitivity of the braid group action on the set of full exceptional sequences of coherent sheaves on a weighted projective line. We do not use here the corresponding result of Crawley-Boevey for modules over hereditary algebras. As an application we prove that the strongest global dimension of the category of coherent sheaves on a weighted projective line $\XX$ does not depend on the parameters of $\XX$. Finally we prove that the determinant of the matrix obtained by taking the values of $n$ $\ZZ$-linear functions defined on the Grothendieck group $K_0(\XX) \simeq \ZZ^n $ of the elements of a full exceptional sequence is an invariant, up to sign.

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Edson R. Alvares, Eduardo N. Marcos, Hagen Meltzer. 2021-02-09. The braid group action on exceptional sequences for weighted projective lines. https://arxiv.org/abs/2102.04584

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