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Holomorphic maps X to Pn correspond to fullest of meromorphic functions
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We claim that there is a bijection between {holomorphic maps from X to Pn } and projective space of dim n over meromorphic functions on X. Given tuple [f_0,...,f_n], we define map phi at every p that is not common zeroes of f_i and not pole of any f_i by phi(p) is just evaluation of the f_i at p. At other points q, choose a different representative of f_i. Conversely, given phi, with x_0 of im phi not identically Zero, take f_0 = 1, f_i = x_i/x_0 composed with phi. Uniqueness: if g_i also corresponds to phi then at every point p where both maps are defined g_i = h(p) f_i for some h. So h is meromorphic.
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