Rational normal curve

A rational normal curve is a smooth rational curve of n in an n- dimensional projective space $\mathbb {P} ^{n}.$ ${\ displaystyle \ mathbb {P} ^ {n}.}$ ${\ mathbb P} ^ {n}.$ It is one of the relatively simple projective varieties , more formally, it is the image of Veronese embedding applied to the projective line.

Content

Definition

A rational normal curve can be defined parametrically as a display image

\nu :\mathbb {P} ^{1}\to \mathbb {P} ^{n}

{\ displaystyle \ nu: \ mathbb {P} ^ {1} \ to \ mathbb {P} ^ {n}}

which translates a point with uniform coordinates $[s:t]$ ${\ displaystyle [s: t]}$ exactly

[s^{n}:s^{n-1}t:s^{n-2}t^{2}:\ldots :t^{n}].

{\ displaystyle [s ^ {n}: s ^ {n-1} t: s ^ {n-2} t ^ {2}: \ ldots: t ^ {n}].}

In affinity map $x_{0}=1$ ${\ displaystyle x_ {0} = 1}$ this mapping is written in a simpler way:

\nu :x\mapsto (x,x^{2},\ldots ,x^{n}).

{\ displaystyle \ nu: x \ mapsto (x, x ^ {2}, \ ldots, x ^ {n}).}

It is easy to see that a rational normal curve is obtained by closing the affine curve $(x,x^{2},\dots ,x^{n})$ ${\ displaystyle (x, x ^ {2}, \ dots, x ^ {n})}$ using a single .

Equivalently, a rational normal curve can be defined as the set of common zeros of homogeneous polynomials

F_{i,j}(x_{0},\ldots ,x_{n})=x_{i}x_{j}-x_{i+1}x_{j-1},

{\ displaystyle F_ {i, j} (x_ {0}, \ ldots, x_ {n}) = x_ {i} x_ {j} -x_ {i + 1} x_ {j-1},}

Where $[x_{0}:\ldots :x_{n}]$ ${\ displaystyle [x_ {0}: \ ldots: x_ {n}]}$ - homogeneous coordinates on $\mathbb {P} ^{n}$ ${\ displaystyle \ mathbb {P} ^ {n}}$ . It is not necessary to consider all these polynomials; to define a curve, it suffices to choose, for example, $F_{i,i}$ ${\ displaystyle F_ {i, i}}$ and $F_{1,n-1}.$ ${\ displaystyle F_ {1, n-1}.}$

Alternative parameterization

Let be $[a_{i}:b_{i}]$ ${\ displaystyle [a_ {i}: b_ {i}]}$ - $n+1$ ${\ displaystyle n + 1}$ different points on $\mathbb {P} ^{1}.$ ${\ displaystyle \ mathbb {P} ^ {1}.}$ Then the polynomial

G(s,t)=\Pi _{i=0}^{n}(a_{i}s-b_{i}t)

{\ displaystyle G (s, t) = \ Pi _ {i = 0} ^ {n} (a_ {i} s-b_ {i} t)}

is a homogeneous polynomial of degree $n+1$ {\ displaystyle n + 1} with different roots. Polynomials

H_{i}(s,t)={\frac {G(s,t)}{(a_{i}s-b_{i}t)}}

{\ displaystyle H_ {i} (s, t) = {\ frac {G (s, t)} {(a_ {i} s-b_ {i} t)}}}

form the basis of the space of homogeneous polynomials of degree n . Display

[s:t]\mapsto [H_{0}(s,t):H_{1}(s,t):\ldots :H_{n}(s,t)]

{\ displaystyle [s: t] \ mapsto [H_ {0} (s, t): H_ {1} (s, t): \ ldots: H_ {n} (s, t)]}

also defines a rational normal curve. Indeed, monomials $s^{n},s^{n-1}t,s^{n-2}t^{2},\ldots ,t^{n}$ ${\ displaystyle s ^ {n}, s ^ {n-1} t, s ^ {n-2} t ^ {2}, \ ldots, t ^ {n}}$ are just one of the possible bases in the space of homogeneous polynomials, and it can be translated by a linear transformation into any other basis.

This mapping sends the zeros of the polynomial $G(s,t)$ ${\ displaystyle G (s, t)}$ to “coordinate points,” that is, points whose all homogeneous coordinates, except one, are equal to zero. Conversely, a rational normal curve passing through these points can be defined parametrically using some polynomial $G .$ ${\ displaystyle G.}$

Properties

Any $n+1$ ${\ displaystyle n + 1}$ points on a rational normal curve in $\mathbb {P} ^{n}$ ${\ displaystyle \ mathbb {P} ^ {n}}$ linearly independent . Conversely, any curve with this property is rational normal.
For any $n+3$ ${\ displaystyle n + 3}$ points in $\mathbb {P} ^{n},$ ${\ displaystyle \ mathbb {P} ^ {n},}$ such that any $n+1$ ${\ displaystyle n + 1}$ of these are linearly independent, there is a single rational normal curve passing through these points. To construct such a curve, it is enough to translate $n+1$ ${\ displaystyle n + 1}$ from points to “coordinate”, and then, if the remaining points have moved to $[c_{0}:c_{1}:\ldots :c_{n}],[d_{0}:d_{1}:\ldots :d_{n}],$ ${\ displaystyle [c_ {0}: c_ {1}: \ ldots: c_ {n}], [d_ {0}: d_ {1}: \ ldots: d_ {n}],}$ as a polynomial $G$ ${\ displaystyle G}$ choose a polynomial that takes place at points $[a_{i}:b_{1}]=[c_{i}^{-1}:d_{i}^{-1}].$ ${\ displaystyle [a_ {i}: b_ {1}] = [c_ {i} ^ {- 1}: d_ {i} ^ {- 1}].}$
Rational normal curve in the case of $n>2$ ${\ displaystyle n> 2}$ is not a complete intersection, that is, it cannot be defined by the number of equations equal to its codimension . ^[one]

Notes

↑ Ravi Vakil . MATH 216: FOUNDATIONS OF ALGEBRAIC GEOMETRY , page 482.

Literature

Harris, J. Algebraic geometry. Beginner course. - M .: ICMMO, 2005 .-- 400 p. - ISBN 5-94057-084-4 .

[1] Ravi Vakil . MATH 216: FOUNDATIONS OF ALGEBRAIC GEOMETRY , page 482.