Composition algebras over algebraic curves of genus zero

We rephrase the classical theory of composition algebras over fields, particularly the Cayley-Dickson Doubling Process and Zorn’s Vector Matrices, in the setting of locally ringed spaces. Fixing an arbitrary base field, we use these constructions to classify composition algebras over (complete smooth) curves of genus zero. Applications are given to composition algebras over function fields of genus zero and polynomial rings.