On the Braid Group Action on Exceptional Sequences for Weighted Projective Lines

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 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 \(\mathbb {X}\) does not depend on the parameters of \(\mathbb {X}\). Finally we prove that the determinant of the matrix obtained by taking the values of n \(\mathbb {Z}\)-linear functions defined on the Grothendieck group \(\textrm{K}_0(\mathbb {X}) \simeq \mathbb {Z}^n \) of the elements of a full exceptional sequence is an invariant, up to sign.