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Let $S$ be the coordinate ring $\mathbb{C}[X_0,...,X_n]$ in $n$-variables. Let $X=\mathrm{Proj}(S/I)$ be a projective scheme where $I$ is an ideal in $S$. Is there a $1-1$ correspondence of first order infinitesimal deformations of $X$ in $\mathbb{P}^n$ (which is isomorphic to $H^0(\mathrm{N}_{X|\mathbb{P}^3})$) and the set of ideals $I'$ in $S \otimes_{\mathbb{C}} \mathbb{C}[t]/(t^2)$ such that $S[t]/(t^2)/I'$ is flat over $\mathbb{C}[t]/(t^2)$ and the image of $I'$ in $S$ (which maps $t$ to zero) is $I$? We know that this is true in the affine case as seen in Hartshorne's "Deformation theory".

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