About projective varieties

Question

Let $f∈Κ[x_0,…,x_n]$ be a homogeneous polynomial. we define $ V(f)={p∈P^n (Κ):f(p)=0}$ is a well-defined subset of $P^n (Κ)$. what is the meaning of well-defined ?

Answer 1

Given a point $[p]\in \mathbb{P}^n(K)$, you would like to define what it means for $f$ to vanish at $[p]$. For this you choose a representative $p$ of $[p]$ and define that $f$ vanishes at $[p]$ if $f(p)=0$. You made a choice in this definition. This definition being well-defined means that it does not depend on the choice made.