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Rational and Elliptic Normal Curve
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Let X be Riemann sphere. Let D be n.infty. Then D is very ample and define a map phi to P^n. ItS image is called a rational normal curve. Let z be coordinate on the finite part of X, then a basis for L(D) is 1, z,.., z^n. The map given by this basis maps point of infty to [0:0:...:1]. When n= 1, this is the standard isom between Riemann sphere and P^1. For n = 2, image is XZ= Y^2. n= 3 corresponds to twisted cubic. For any choice of basis of L(n.infty), i.e, polynomial of degree at most n, the resulting map is called a veronese map. Let X be complex Torus. Then any divisor of degree m at least 3 is very ample and thus gives a map to P^{m-1}. Its image is called an elliptic normal curve of degree d.
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