Thm 1.4 States that If $\Gamma\subset P^n$ is any collection of $d\leq 2n$ points in general position, then $\Gamma$ can be described by quadratic polynomials $\{f_i\}$.
The proof intends to show that for any $q$ such that if for all i, $f_i(\Gamma)=f(q)=0$, then $q\in\Gamma$ and it turns out that $q$ lies in a set of cardinality $|n|$. And then it wants to show that $q$ is an element of that set. So one took advantage of redundant information of 2n points in general position and $q$ lies in a set of cardinality $|n|$ to finish the proof.
What is the geometrical picture that forces $q$ to be an element of that special set?