Let C be a projective curve, P ∈C. Then there is a birational morphism f: C →C^', C^' a projective plane curve, such that f^-1(f(P))={P}. We outline a proof: (a) We can assume: C ⊂ℙ^n 1 Let T, X1, …, Xn, Z be coordinates for ℙ^n 1; Then C ∩V(T) is finite; C ∩V(T, Z)=∅; P=[0: …: 0: 1]; and k(C) is algebraic over k(u), where u=T̅ / Z̅ ∈k(C). (b) For each λ=(λ1, …, ) ∈k^n, let φλ: C →ℙ^2 be defined by the formula φ(|t : ..x1: …: xn: z])=[t: ∑xi: z]. Then φλis a well-defined morphism, and φλ(P)= [0:

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