The intuition behind the coordinate ring $\Gamma(F)$

Question

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

Answer 1

We want the coordinate ring to be the ring of all polynomial functions on V. Two (polynomial) functions will be the same on V if they have the same values at every point of V (they may be different outside of V). This is the same as saying their difference is 0 all across V. So we want to mod out by all of these polynomial functions and I(V) is by definition all the polynomials that vanish on all of V. So now, two distinct elements in the coordinate ring are actually distinct functions on V.