I'm reading some notes on hodge theory by Charles Siegel which makes a claim on page 16 relating the space of deformations of a smooth projective hypersurface $X$ with the jacobian ideal. More specifically, let
$$
Proj(\mathbb{C}[x\_1,\ldots, x\_n]/(f)) = Proj(S\_\bullet) = X
$$
then
$$
H^1(X,T\_X) = \frac{S\_d}{\text{Jac}(f)\_d}
$$
where can I find a proof of this and are there any generalizations?



Where can I find a proof of identity of $H^1(X,T_X)$ and a quotient by the jacobian?
This is just a standard application of Griffiths's Residue theory. 
For a complete treatment you may consult Claire Voisin's book no.2 on Hodge Theory, but let me just give you the idea.


In general, under some mild hypotheses, the whole Jacobian ring $R=S/Jac(f)$ describes the (embedded) deformations of the affine cone $A\_X$ over $X$.
Morally speaking you have to imagine that an element $g \in R\_k$ gives rise to a deformation given by, say, $A\_{\epsilon}= V(f+\epsilon g)$.


With this in mind, is not difficult for an hypersurface to convince yourself that $R\_d \cong H^1(X, T\_X)$ (just think to what deformations move the affine cone over $X$ to another affine cone over an $X'$).


Actually what you can prove is that $R\_k \cong H^1(X, T\_X(-d+k))$. To do this you might either try tangent/normal sequence as said above or write down the sequence associated to $c\_1(\mathcal{O}\_X(1)) \in$ Ext$^1(T\_X, \mathcal{O}\_X)$ (the so-called Atiyah extension).


