Finite algebraic variety of projective space

Question

I came across this theorem (which I didn't understand) in Algebraic geometry by Harris. Suppose $F$ is a subset of $\mathbb{P}^n$ (projective space of dimension $n$) containing $d<2n$ points in general position. Then $F$ can be described as the zero locus of quadratic polynomials. I am new at algebraic geometry so any clear step by step argument is appreciated.

Answer 1

This is theorem 1.4 in Harris. The term general position is defined immediately above the theorem.

Now, by assumption $q$ is a point such that every quadratic polynomial $f$ which vanishes at each point of $\Gamma$ also vanishes at $q$. Since $\Gamma$ consists of $2n$ points in general position, the first $n$ points span a hyperplane $\Lambda_1$ and the other $n$ points span a hyperplane $\Lambda_2$.

Write $f_1$ for the linear polynomial cutting out $\Lambda_1$ and $f_2$ for the linear polynomial cutting out $\Lambda_2$. Then $f = f_1 f_2$ is a quadratic polynomial which vanishes precisely on $\Lambda_1$ and $\Lambda_2$. In particular, $f$ vanishes at each point of $\Gamma$. By assumption, this means $f$ also vanishes at $q$. So it follows that $q$ lies on either $\Lambda_1$ or $\Lambda_2$.