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 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 X does not depend on the parameters of X. Finally we prove that the determinant of the matrix obtained by taking the values of n Z-linear functions defined on the Grothendieck group K₀(X) ≃ Zⁿ of the elements of a full exceptional sequence is an invariant, up to sign.