I'm reading this paper, and in page $3$ is calculated the Hilbert Function for an surface $S$ with degree $d\ge 4$ in $\mathbb P^3$. I' ve tried to use the method described here, but I can't get the same result. Any clue in how to compute?
Thanks in advance.
I'm pretty rusty, sorry if I'm missing something. In particular I don't see how degree $\ge 4$ matters.
The ring of regular functions is given by $R/(f)$ where $R=\mathbb{k}[x_0,x_1,x_2,x_3]$ and $f$ has homogeneous degree $d$. Looking at the degree $x$ part of this quotient, it's the quotient $$ R_{x}/(R_{x-d}f) $$
and $\dim(R_x) = \binom{x+3}{3}$ and $\dim(R_{x-d})=\binom{x-d+3}{3}$