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54321user
54321user
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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?

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 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?

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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54321user
54321user
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Where can I find a proof of identity of $H^1(X,T_X)$ and a quotient by the jacobian?

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 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?