Twisted Cubic

# Twisted cubic

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Description:
In mathematics, a twisted cubic is a smooth, rational curve C of degree three in projective 3-space P<sup>3</sup>. 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 the image of a Veronese map of degree three on the projective line.

## Definition

It is most easily given parametrically as the image of the map

[itex]nu:mathbf^1tomathbf^3[/itex]

which assigns to the homogeneous coordinate [itex][/itex] the value

[itex]nu: mapsto .[/itex]

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

[itex]nu:x mapsto (x,x^2,x^3)[/itex]

That is, it is the closure by a single point at infinity of the affine curve [itex](x,x^2,x^3)[/itex].

Equivalently, it is a projective variety, defined as the zero locus of three smooth quadrics. Given the homogeneous coordinates on P<sup>3</sup>, it is the zero locus of the three homogeneous polynomials

[itex]F_0 = XZ - Y^2[/itex]
[itex]F_1 = YW - Z^2[/itex]
[itex]F_2 = XW - YZ.[/itex]

It may be checked that these three quadratic forms vanish identically when using the explicit...

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