Twisted cubic

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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

<math>nu:mathbf^1tomathbf^3</math>


which assigns to the homogeneous coordinate <math></math> the value

<math>nu: mapsto .</math>


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

<math>nu:x mapsto (x,x^2,x^3)</math>


That is, it is the closure by a single point at infinity of the affine curve <math>(x,x^2,x^3)</math>.

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

<math>F_0 = XZ - Y^2</math>
<math>F_1 = YW - Z^2</math>
<math>F_2 = XW - YZ.</math>


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