A complex line bundle is a vector bundle π:E->M whose fibers π^(-1)(m) are a copy of C. π is a holomorphic line bundle if it is a holomorphic map between complex manifolds and its transition functions are holomorphic. On a compact Riemann surface, a variety divisor sum n_i p_i determines a line bundle. For example, consider 2p - q on X. Around p there is a coordinate chart U given by the holomorphic function z_p with z_p(p) = 0. Similarly, z_q is a holomorphic function defining a disjoint chart V around q with z_q(q) = 0. Then letting W = X - {p, q}, the Riemann surface is covered by X = U union V union W.
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