Twisted cubic

From Wikipedia, the free encyclopedia

In mathematics, a twisted cubic is a smooth, rational curve $C$ of degree three in projective 3-space $\mathbb{P}^3$ . It is a fundamental example of a skew curve. It is essentially unique, up to projective transformation (the twisted cubic, therefore.) It is generally considered to be the simplest example of a projective variety that isn't linear or a hypersurface, and is given as such in most textbooks on algebraic geometry. It is the three-dimensional case of the rational normal curve, and is a Veronese map of degree three.

[edit] Definition

It is most easily given parametrically as the map

$\nu:\mathbb{P}^1\to\mathbb{P}^3$

which assigns to the homogeneous coordinate $[S : T]$ the value

$\nu:[S:T] \mapsto [S^3:S^2T:ST^2:T^3]$

In the affine coordinates of projective space, the map is simply

$\nu:x \mapsto (x,x^2,x^3)$

That is, it is the closure by a single point at infinity of the affine curve $(x, x 2, x 3)$ .

Equivalently, it is a projective variety, defined as the zero locus of three smooth quadrics. Given the homogeneous coordinates $[X : Y : Z : W]$ on $\mathbb{P}^3$ , it is the zero locus of the three homogeneous polynomials

F 0 = X Z - Y 2

F 1 = Y W - Z 2

F 2 = X W - Y Z

It may be checked that these three quadratic forms vanish identically when using the explicit parameterization above; that is, substituting $S 3$ for $X$ , and so on.

Equivalently, the homogeneous ideal of the twisted cubic $C$ is generated by three algebraic forms of degree two on $\mathbb{P}^3$ . The generators of the ideal are

{X Z - Y 2, Y W - Z 2, X W - Y Z}

[edit] Properties

The twisted cubic has an assortment of curious properties:

Any four points on $C$ span $\mathbb{P}^3$ .
Given six points in $\mathbb{P}^3$ with no four coplanar, there is a unique twisted cubic passing through them.
The tangent and secant lines of a twisted cubic $C$ fill up $\mathbb{P}^3$ and are pairwise disjoint, except at points of the curve itself.
The projection of $C$ onto a plane from a point on a tangent line of $C$ yields a cuspidal cubic.
The projection from a point on a secant line of $C$ yields a nodal cubic.
The projection from a point on $C$ yields a conic section.

[edit] References

Joe Harris, Algebraic Geometry, A First Course, (1992) Springer-Verlag, New York. ISBN 0-387-97716-3

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Category: Algebraic curves

Twisted cubic

From Wikipedia, the free encyclopedia

[edit] Definition

[edit] Properties

[edit] References

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