Properties
The rational normal curve has an assortment of nice properties:
- Any points on are linearly independent, and span . This property distinguishes the rational normal curve from all other curves.
- Given points in in linear general position (that is, with no lying in a hyperplane), there is a unique rational normal curve passing through them. The curve may be explicitly specified using the parametric representation, by arranging of the points to lie on the coordinate axes, and then mapping the other two points to and .
- The tangent and secant lines of a rational normal curve are pairwise disjoint, except at points of the curve itself. This is a property shared by sufficiently positive embeddings of any projective variety.
There are independent quadrics that generate the ideal of the curve.
The curve is not a complete intersection, for . This means it is not defined by the number of equations equal to its codimension .
The canonical mapping for a hyperelliptic curve has image a rational normal curve, and is 2-to-1.
Every irreducible non-degenerate curve of degree is a rational normal curve.
Read more about this topic: Rational Normal Curve
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—John Locke (1632–1704)
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