Twisted cubic

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In mathematics, a twisted cubic is a smooth, rational curve C of degree three in projective 3-space P³. It is a fundamental example of a skew curve. It is essentially unique, up to projective transformation (the twisted cubic, therefore). In algebraic geometry, the twisted cubic is a simple example of a projective variety that is not linear or a hypersurface, in fact not a complete intersection. It is the three-dimensional case of the rational normal curve, and is the image of a Veronese map of degree three on the projective line.

Definition

The twisted cubic is most easily given parametrically as the image of the map

\nu :\mathbf {P} ^{1}\to \mathbf {P} ^{3}

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

\nu :[S:T]\mapsto [S^{3}:S^{2}T:ST^{2}:T^{3}].

In one coordinate patch of projective space, the map is simply the moment curve

\nu :x\mapsto (x,x^{2},x^{3})

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

The twisted cubic is a projective variety, defined as the intersection of three quadrics. In homogeneous coordinates $[X:Y:Z:W]$ on P³, the twisted cubic is the closed subscheme defined by the vanishing of the three homogeneous polynomials

F_{0}=XZ-Y^{2}

F_{1}=YW-Z^{2}

F_{2}=XW-YZ.

It may be checked that these three quadratic forms vanish identically when using the explicit parameterization above; that is, substitute x³ for X, and so on.

More strongly, the homogeneous ideal of the twisted cubic C is generated by these three homogeneous polynomials of degree 2.

Properties

The twisted cubic has the following properties:

It is the set-theoretic complete intersection of $XZ-Y^{2}$ and $Z(YW-Z^{2})-W(XW-YZ)$ , but not a scheme-theoretic or ideal-theoretic complete intersection; meaning to say that the ideal of the variety cannot be generated by only 2 polynomials; a minimum of 3 are needed. (An attempt to use only two polynomials make the resulting ideal not radical, since $(YW-Z^{2})^{2}$ is in it, but $YW-Z^{2}$ is not).
Any four points on C span P³.
Given six points in P³ with no four coplanar, there is a unique twisted cubic passing through them.
The union of the tangent and secant lines (the secant variety) of a twisted cubic C fill up P³ and the lines are pairwise disjoint, except at points of the curve itself. In fact, the union of the tangent and secant lines of any non-planar smooth algebraic curve is three-dimensional. Further, any smooth algebraic variety with the property that every length four subscheme spans P³ has the property that the tangent and secant lines are pairwise disjoint, except at points of the variety 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.

References

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

Topics in algebraic curves

Rational curves

Elliptic curves

Analytic theory	Elliptic function Elliptic integral Fundamental pair of periods Modular form
Arithmetic theory	Counting points on elliptic curves Division polynomials Hasse's theorem on elliptic curves Mazur's torsion theorem Modular elliptic curve Modularity theorem Mordell–Weil theorem Nagell–Lutz theorem Supersingular elliptic curve Schoof's algorithm Schoof–Elkies–Atkin algorithm
Applications	Elliptic curve cryptography Elliptic curve primality

Higher genus

Plane curves

Riemann surfaces

Constructions

Structure of curves

Divisors on curves	Abel–Jacobi map Brill–Noether theory Clifford's theorem on special divisors Gonality of an algebraic curve Jacobian variety Riemann–Roch theorem Weierstrass point Weil reciprocity law
Moduli	ELSV formula Gromov–Witten invariant Hodge bundle Moduli of algebraic curves Stable curve
Morphisms	Hasse–Witt matrix Riemann–Hurwitz formula Prym variety Weber's theorem (Algebraic curves)
Singularities	A_k singularity Acnode Crunode Cusp Delta invariant Tacnode
Vector bundles	Birkhoff–Grothendieck theorem Stable vector bundle Vector bundles on algebraic curves

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