Consider the Veronese map $v_2,_2$: $\mathbb{P^2}$ $\Rightarrow$ $\mathbb{P^5}$. I'm asked to show that all lines in $\mathbb{P^2}$ are mapped into conics contained in some subspace M of $\mathbb{P^5}$.
I get that M is a $\mathbb{P^2}$$\subset$$\mathbb{P^5}$, so the image of a line L is some plane curve. How do I formalize it? And then, how do I say that the image is actually a conic, and not a cubic or something else?
Any hint or solution sketch would be great :)
HINT:
Let $p=x^2$, $q=y^2$, $r=z^2$, $u=y z$, $v=x z$, $w=x y$. Consider the equation of a line $a x + b y + c z = 0$. We conclude the following equalities: $$a p + b w + c v = 0\\ a w + b q + c u = 0\\ a v + b u + c r = 0\\ a v w + b u w + c u v =0$$ Note that we can change the system of coordinates of $\mathbb{P}^2$ so that $a$, $b$, $c$ are all non-zero.