Let C be a decomposable plane curve over an algebraically closed field k of characteristic 0. That is, C is defined in k^2 by an equation of the form g(x) = f(y), where g and f are polynomials of degree at least 2. We use this data to construct 3 pointed Hopf algebras, A(x, a, g), A(y, b, f) and A(g, f), in the first two of which g [resp. f] are skew primitive central elements, and the third being a factor of the tensor product of the first two. We conjecture that A(g, f) contains the coordinate ring O(C) of C as a quantum homogeneous space, and prove this when each of g and f has degree at most 5 or is a power of the variable. We obtain many properties of these Hopf algebras, and show that, for small degrees, they are related to previously known algebras. For example, when g has degree 3, A(x, a, g) is a PBW deformation of the localisation at powers of a generator of the down-up algebra A(−1, −1, 0).

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