In
mathematics, the rational normal curve is a smooth,
rational curve C of
degree n in
projective n-space Pn. It is a simple example of a
projective variety; formally, it is the
Veronese variety when the domain is the projective line. For n = 2 it is the
plane conic Z0Z2 = Z2
1, and for n = 3 it is the
twisted cubic. The term "normal" refers to
projective normality, not
normal schemes. The intersection of the rational normal curve with an
affine space is called the
moment curve.
Definition
The rational normal curve may be given
parametrically as the image of the map
which assigns to the
homogeneous coordinates S : T the value
In the
affine coordinates of the chart x0 ≠ 0 the map is simply
That is, the rational normal curve is the closure by a single
point at infinity of the
affine curve
Equivalently, rational normal curve may be understood to be a
projective variety, defined as the common zero locus of the
homogeneous polynomials
where are the
homogeneous coordinates on Pn. The full set of these polynomials is not needed; it is sufficient to pick n of these to specify the curve.
Alternate parameterization
Let be n + 1 distinct points in P1. Then the polynomial
is a
homogeneous polynomial of degree n + 1 with distinct roots. The polynomials
are then a
basis for the space of homogeneous polynomials of degree n. The map
or, equivalently, dividing by G(S, T)
is a rational normal curve. That this is a rational normal curve may be understood by noting that the
monomials
are just one possible
basis for the space of degree n homogeneous polynomials. In fact, any
basis will do. This is just an application of the statement that any two projective varieties are projectively equivalent if they are
congruent modulo the
projective linear group PGLn + 1(K) (with K the
field over which the projective space is defined).
This rational curve sends the zeros of G to each of the coordinate points of Pn; that is, all but one of the Hi vanish for a zero of G. Conversely, any rational normal curve passing through the n + 1 coordinate points may be written parametrically in this way.
Properties
The rational normal curve has an assortment of nice properties:
- Any n + 1 points on C are linearly independent, and span Pn. This property distinguishes the rational normal curve from all other curves.
- Given n + 3 points in Pn in linear
general position (that is, with no n + 1 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 n + 1 of the points to lie on the coordinate axes, and then mapping the other two points to S : T] = [0 : 1] and S : T] = [1 : 0].
- 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 n > 2. That is, it cannot be defined (as a
subscheme of projective space) by only n − 1 equations, that being the
codimension of the curve in .
- The
canonical mapping for a
hyperelliptic curve has image a rational normal curve, and is 2-to-1.
- Every irreducible non-degenerate curve C ⊂ Pn of degree n is a rational normal curve.
See also
References
- Joe Harris, Algebraic Geometry, A First Course, (1992) Springer-Verlag, New York.
ISBN
0-387-97716-3