Twisted cubic

The twisted cubic is the image ofin ${\mathbb{P}}^3$. Its ideal is generated by the three $2\times 2$ minors ofand it has degree three.

Modulo the three quadric relations of the twisted cubic, replace $x_1^2$ by $x_0{x}_2$, $x_2^2$ by $x_1{x}_3$, and $x_1{x}_2$ by $x_0{x}_3$. Each replacement reduces the total exponent of $x_1,x_2$. The resulting monomials are $x_0^a{x}_3^b$, $x_0^a{x}_1{x}_3^b$, and $x_0^a{x}_2{x}_3^b$. In degree $m$ their images under $x_i\mapsto s^{3-i}{t}^i$ are all $3m+1$ distinct monomials of degree $3m$. They are linearly independent, so the three quadrics generate the full homogeneous ideal, not merely its radical. The quotient is the third Veronese subring of $k[s,t]$.

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 ${\mathbb{P}}^3$, making their degree product at least four rather than the curve's degree three.