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".


This site is temporarily in read only mode and not accepting new answers.

Browse other questions tagged .