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 (DX)-X-b of coherent sheaves on a weighted projective line X, equivalently 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 s emi-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.