I'm studying Fulton's algebraic curves book. He gives the following definitions:
We can define the coordinate ring of a nonempty variety $V\subset \mathbb A^n$ as $\Gamma(V)=k[X_1,\ldots,X_n]/I(V)$.
When $V=V(F)$, where $F$ is irreducible curve, we use this notation: $\Gamma(F)\doteqdot \Gamma(V(F))$.
I'm trying to understand intuitively the coordinate ring $\Gamma(F)$, I know the elements of $\Gamma(F)$ are the polynomial functions $h:V(F)\to k$. Then, every point in the curve $F$, we can associate a value in $k$. Is there some geometric intuition behind this fact?
Thanks