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Consider the map $\phi : \mathbb P_k^1 \to \mathbb P_k^3$, where $ \phi(t_0:t_1) = (t_0^3 : t_0^2 t_1:t_0t_1^2:t_1^3)$. Apparently, the image $C := \phi(\mathbb P_k^1)$ is a projective variety given by $C = \left\{ (x_0 : x_1: x_2: x_3) \in \mathbb P_k^3 \ | \ \mathrm{rank} \begin{pmatrix} x_0 & x_1 & x _2 \\ x_1 & x_2 & x_3 \end{pmatrix} \leq 1 \right\}$.

Why is this so? How does one arrive at such a description? I can see why such a description of $C$ shows that it is a projective variety (the statement with rank is equivalent to three quadric equations), but I don't see where the expression comes from.

Thanks.

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It's clear that the image is determined by the equations $x_0x_2 - x_1^2 = 0, \ x_0x_3 - x_1x_2 = 0$ and $x_1 x_3 - x_2^2 = 0$. These three equations are equivalent to $\mathrm{rank}\begin{pmatrix} x_0 & x_1 & x_2 \\ x_1 & x_2 & x_3 \end{pmatrix} \leq 1$, but this is entirely unsatisfying. What's the reason 'why' we can express this variety in terms of the rank of some matrix? –  Matt Mar 7 '12 at 11:30
    
This reminds me of the Plücker embedding. –  Martin Brandenburg Mar 9 '12 at 13:36
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