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Let f_0, …, f_n be a basis of L(D). Let phi: X to P^n be induced by f_i. Thus for p in X, if g is a meromorphic function st ord_p g = min ord_p f_i then phi(p) = [f_0/g(p), …, f_n/g(p)]. Identify P^n with P(L(D)*) by indetifying [0,…,1,…] with f_i*. Then phi(p) corresponds to the linear functional (sum f_i/g (p) f_i^*). Its kernel is the codimension 1 subspace of P(L(D)) consisting of f = sum a_i f_i such that sum a_i f_0/g(p) = 0. This means that f/g (p) = 0, and thus ord_p f - ord_p g is at least 1. So ord_p f is at least -D(p) +1. Thus f is in P(L(D-p)). As a map from X to |D|* (space of codimension 1 subspaces of |D|), phi sends p to {E in |D| s.t. E is geq p}
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