Exceptional sequences determined by their Cartan matrix

We investigate complete exceptional sequences E = (E₁,…, E_n) in the derived category D^b ∧ of finite-dimensional modules over a canonical algebra, equivalently in the derived category D^b (X double struck) of coherent sheaves on a weighted projective line, and the associated Cartan matrices C(E) = (〈[E_i], [E_j]〉). As a consequence of the transitivity of the braid group action on such sequences we show that a given Cartan matrix has at most finitely many realizations by an exceptional sequence E, up to an automorphism and a multi-translation (E₁,…, E_n)→ (E₁ [i₁],…, E_n [i_n]) of D^b ∧. Moreover, we determine a bound on the number of such realizations. Our results imply that a derived canonical algebra A is determined by its Cartan matrix up to isomorphism if and only if the Hochschild cohomology of A vanishes in nonzero degree, a condition satisfied if A is representation-finite.