In page : $206$ of the following electronic textbook : https://scholar.harvard.edu/files/joeharris/files/000-final-3264.pdf , the author says that the Hilbert function of $ S_X $ is : $$ \mathrm{dim} \ ( S_X )_d = \begin{pmatrix} n+d \\ n \end{pmatrix} - \begin{pmatrix} n+d-s \\ n \end{pmatrix} $$ with, $ X $ is any hypersurface of degree $s$ in $ \mathbb{P}^n $.
Can you explain to me please, what is $ S_X $, and how do we obtain this formula above ?
Thanks in advance for your help.
$S_X$ is the homogeneous co-ordinate ring of $X$, which in this case is just $S/F$, where $S$ the homogeneous co-ordinate ring of the projective space, which is just a polynomial ring in $n+1$ variables and let $S_m$ denote the vector space of homogeneous polynomials of degree $m$ in these $n+1$ variables. For any $d\geq 0$, one has an exact sequence, $0\to S_{d-s}\stackrel{F}{\to} S_d\to (S_X)_d\to 0$. Thus the dimension of interest is $\dim S_d-\dim S_{d-s}$. The choose functions are just these dimensions. In other words, $\dim S_m={n+m\choose n}$.
