Suppose $R=F[X_0,\dots,X_r]$, where $F$ is an algebraically closed field. Now $R$ is graded, with the homogeneous polynomials of degree $n$ being the elements of degree $n$. Now suppose $I$ is a homogeneous radical ideal. As usual, we have the functions $\phi(n)=\dim_F R_n$, $\phi(n,I)=\dim_F I_n$, and $\chi(n,I)=\dim_F R_n/I_n=\dim_F R_n-\dim_F I_n=\phi(n)-\phi(n,I)$.
I've been wondering about the following. Suppose $\chi(n,I)=m$ for all large enough $n$. From this, why is it that the zeroes of $I$ in $\mathbb{P}^r$ consist of $m$ distinct points?
Now $\phi(n)=\binom{r+n}{r}$, so as $n$ increases, $\phi(n)$ increases, so necessarily $\phi(n,I)$ must get larger to balance out $\chi(n,I)$. However, I don't see how this relates to the zeroes of $I$ in $\mathbb{P}^r$.