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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

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    $\begingroup$ As a general rule, either define your notation (what is $F$? What is $V(F)$) or tell us at least what book's notation you are using... $\endgroup$
    – Thomas Andrews
    Aug 13, 2014 at 19:42
  • $\begingroup$ @ThomasAndrews Thank you for the remark $\endgroup$
    – user42912
    Aug 13, 2014 at 19:51
  • $\begingroup$ @ThomasAndrews as it ok, now? Thanks again. $\endgroup$
    – user42912
    Aug 13, 2014 at 19:54
  • $\begingroup$ What do you mean by "$V(F)$" when $F$ is a curve? $\endgroup$
    – bradhd
    Aug 13, 2014 at 19:57
  • $\begingroup$ @Brad $V(F)$ is the points which vanishes on $F$. $\endgroup$
    – user42912
    Aug 13, 2014 at 22:01

1 Answer 1

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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.

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