The braid group action on exceptional sequences for weighted projective lines

TL;DR

This paper provides a new proof of the braid group action's transitivity on exceptional sequences in weighted projective lines, and explores invariants related to the category of coherent sheaves, with implications for global dimension.

Contribution

It offers an intrinsic proof of braid group action transitivity and establishes invariants of exceptional sequences independent of parameters.

Findings

Braid group action on exceptional sequences is transitive.

The global dimension of coherent sheaves category is parameter-independent.

Determinant of certain matrices from exceptional sequences is an invariant up to sign.

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.