Let $\nu:P^1 \rightarrow P^2$ be the veronese map of degree $2$, i.e. $[Y_0 : Y_1] \mapsto [Y_0^2 : Y_0 Y_1 : Y_1^2]$ and let $\sigma: P^1 \times P^2 \rightarrow P^5$ be the Segre map. Consider the image $\mathcal{X}$ of $\sigma \circ \nu$, i.e. the image of \begin{align} [X_0:X_1] \times[Y_0:Y_1]\stackrel{\nu}{\mapsto} [X_0:X_1]\times[Y_0^2 : Y_0 Y_1 : Y_1^2]\stackrel{\sigma}{\mapsto}[X_0 Y_0^2: X_0 Y_0 Y_1 : X_0 Y_1^2 : X_1 Y_0^2: X_1 Y_0 Y_1 : X_1 Y_1^2]. \end{align}
Then $\mathcal{X}$ is a projective variety of $P^5$ given by $7$ quadratic equations ($3$ for the Segre variety and $4$ for the image of the veronese variety).
Next, let $C_4$ be the rational normal curve, which arises as the image of $P^1 \rightarrow P^4$ given by \begin{align} [W_0 : W_1] \mapsto [W_0^4: W_0^3 W_1: W_0^2 W_1^2 : W_0 W_1^3 : W_1^4]. \end{align}
$C_4$ is given by $4$ quadratic equations.
Exercise 2.28 in Harris (AG-first course) says that $C_4$ can be realized as a hyperplane section of $\mathcal{X}$. I have been trying to see why this is true, by working with the pullback of $C_4$ under $\sigma$. More precisely, to prove the statement in Harris i would have to show that this pullback is given by the equation $Y_1^2 - Y_0Y_2=0$, which is the veronese variety, together with a bilinear form in $X_1,X_0$ (homogeneous coordinates of $P^1$) and $Y_0,Y_1,Y_2$ (homogeneous coordinates of $P^2$), which would correspond to a hyperplane in $P^5$.
When i pull back the equations of $C_4$ i get:
$(1): \, X_0^2(Y_1^2-Y_0 Y_2) = 0 \\ (2): \, X_0(X_0 Y_2^2 - X_1 Y_0 Y_1)=0 \\ (3): \, X_1(X_1 Y_0^2 - X_0 Y_1 Y_2)=0.$
Issue 1: The variety described by these equations does not entirely live in the veronese variety (since equation (1) is multiplied by $X_0^2$).
Issue 2: I am finding hard to believe that equations (1)-(3) can give the same variety of $P^1 \times P^2$ as the one given by $Y_1^2-Y_0 Y_2=0$ together with a bilinear form.
Question: How can these two issues be addressed?
PS: I will be happy to provide more details upon request; did not do so in the first place to avoid overloading the question. Also, general hints are good enough for me, no need to go into details.
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For convenience i attach the description from Harris: