Today I began reading Hartshorne, and was doing an exercise about the twisted cubic, parametrically given by $t \mapsto (t,t^2,t^3)$. According to Wikipedia, its (non-homogeneous) ideal is $(xz-y^2,y-z^2,x-yz)$. Toying a bit with these equations, I noticed that they are precisely the maximal minors of the matrix
$\begin{pmatrix}x &y &z \\ y& z& 1\end{pmatrix}$
Is there a reason for this? I tend to think of determinants as having to do only with linear equations of some sort, but the twisted cubic does not seem very "linear".