The automorphism group of the derived category for a weighted projective line

We show that up to a translation each automorphism of the derived category D ^b X of coherent sheaves on a weighted projective line X, equiv-alently of the derived category D ^b A of finite dimensional modules over a derived canonical algebra A, is composed of tubular mutations and automorphisms of X. In the case of genus one this implies that the automorphism group is a semi-direct product of the braid group on three strands by a finite group.

Moreover we prove that most automorphisms lift from the Grothendieck group to the derived category.