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Let $X$ be an hypersurface of degree $k$ in $\mathbb{P}^{n}$, why the equation defining $X$ becomes linear in the Veronese coordinates?

More precisely I want to understand the last paragraph of the following link:

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up vote 7 down vote accepted

Let me do an example, which illustrates the general case.

Suppose you have the conic $C\subseteq P^2$ with equation $xy=z^2$. The second Veronese mappoing is $$V_2:(x:y:z)\in P^2\longmapsto(x^2:y^2:z^2:xy:zx:yz)\in P^5.$$ Call $v_0$, $\dots$, $v_5$ the homogeneous coordinates in $P^5$.You should have no trouble showing that the image $V_2(C)$ of $C$ under the map $V_2$ is exactly the set of points of $V_2(P^2)$ which are on the hyperplane with equation $v_3-v_2=0$.

In general, a degree $d$ curve on $P^2$ is mapped under $V$ to a hyperplane section of $V_d(P^2)$, the image under the $d$th Veronese of $P^2$.

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Indeed! ${}{}{}$ – Mariano Suárez-Alvarez May 1 '12 at 6:02
Dear Mariano, $\mathbb P^5$ has no $v_6$ homogeneous coordinate. – Georges Elencwajg May 1 '12 at 7:26
I'll eventually learn that :D (I hope!) – Mariano Suárez-Alvarez May 1 '12 at 8:23

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