How to prove this comment of Fulton

Question

I'm trying to understand why this is true in Fulton's Algebraic Curves:

Why we add this point $(0,\ldots, 0)$? Why this equality is true? I really need help.

Thanks in advance.

Answer 1

Let $f\in\mathbb{K}[x_1,\ldots,x_{n+1}]$ be homogeneous. Then $(x_1,\ldots,x_{n+1})\in C(V_p(f))$ if and only if $[x_1:\ldots:x_{n+1}]\in V_p(f)$ if and only if $f(x_1,\ldots,x_{n+1})=0$. This is it, I think.

Oh, there is also $p=(0,\ldots,0)$, of course, but since $f$ is homogeneous, $p\in V_a(f)$.