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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?

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  • $\begingroup$ $X$ is smooth and $d$ is the degree of $f$? Have you tried smashing the conormal and Euler short exact sequences against this? $\endgroup$ – Hoot Aug 7 '16 at 22:55
  • $\begingroup$ @Hoot I would assume the degree condition would be true, but this is not clear from the notes. I'll try and see what I get from these. $\endgroup$ – 54321user Aug 7 '16 at 23:01
  • $\begingroup$ Yes, $d$ is the degree of $f$, and this is indeed immediate from the two exact sequences mentioned by Hoot. $\endgroup$ – abx Aug 8 '16 at 5:21
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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).

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