The twisted cubic is the image ofin . Its ideal is generated by the three minors ofand it has degree three.
A projective variety is nondegenerate when it is contained in no hyperplane of its ambient projective space.
The twisted cubic is a nondegenerate projective variety, so every hypersurface containing it has degree at least two. At least two hypersurfaces are needed to cut out a curve in , making their degree product at least four rather than the curve's degree three.
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The twisted cubic is a specific type of space curve in three-dimensional space, often discussed in the context of algebraic geometry and differential geometry. It can be defined parametrically as a function of a single variable \( t \): \[ \mathbf{c}(t) = (t, t^2, t^3) \] for \( t \) in the real numbers, \( \mathbb{R} \).