  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.
