Let $ k $ be an algebraically closed field.
Consider the map $$ \varphi : \mathbb{P}^{1}_{k} \rightarrow \mathbb{P}^{3}_{k}, $$ $$ (t_{0}:t_{1}) \mapsto (t^{3}_{0}:t^{2}_{0}t_{1}:t_{0}t^{2}_{1}:t^{3}_{1}). $$
According to Klaus Hulek's Elementary Algebraic Geometry, $ C:= \varphi(\mathbb{P}^{1}_{k}) $ is a projective variety given by
$$ C = \Bigg\lbrace (x_{0}:x_{1}:x_{2}:x_{3}) \in \mathbb{P}^{3}_{k} \Bigg|\ \text{rank}\begin{pmatrix} x_{0} & x_{1} & x_{2} \\ x_{1} & x_{2} & x_{3} \end{pmatrix} \leq 1 \Bigg\rbrace $$
My first question is why this is the image of $ \varphi. $
Furthermore, the author notes that $ C = Q_{1} \cap Q_{2} \cap Q_{3}, $ where $$ Q_{1} = \lbrace (x_{0}:x_{1}:x_{2}) \in \mathbb{P}^{2}_{k}| \; x_{0}x_{2}-x^{2}_{1} = 0 \rbrace $$ $$ Q_{2} = \lbrace (x_{0}:x_{1}:x_{2}) \in \mathbb{P}^{2}_{k}| \; x_{0}x_{3}-x_{1}x_{2} = 0 \rbrace $$ $$ Q_{3} = \lbrace (x_{0}:x_{1}:x_{2}) \in \mathbb{P}^{2}_{k}| \; x_{1}x_{3}-x^{2}_{2} = 0 \rbrace. $$
I notice the defining polynomial of each of these quadrics is the determinant of a minor of the matrix above, and I don't know why. This may be an obvious thing, but it is quite interesting to me.