Let $X$ be a smooth complex projective variety. Suppose $X \hookrightarrow \mathbb{P}^n$. Let $Z$ be a closed (reduced) subscheme of $X$. Let $X'$ be an infinitesimal deformation of $X$ corresponding to a global section $s \in H^0(\mathcal{N}_{X|\mathbb{P}^n})$. By this we mean that there exists a flat morphism $f:X' \to \mathbb{C}[t]/(t^2)$ such that $X \cong X' \times_{\mathbb{C}[t]/(t^2)} \mathbb{C}$. Denote by $i:X \to X'$ the natural morphism arising from the definition of infinitesimal deformation of $X$. Suppose that the image of $s$ in $(\mathcal{N}_{X|\mathbb{P}^n})|_Z$ is equal to zero. Does this imply that there exists an open set $U \subset X'$ containing $i(Z)$ such that $U$ is isomorphic to $i^{-1}(U) \times \mathbb{C}[t]/(t^2)$ i.e, $s$ induces a trivial deformation on an open neighbourhood containing $Z$?

  • $\begingroup$ I would think this follows directly from the association of $X'$ to an element in $H^0(\mathcal{N}_{X|\mathbb{P}^n})$ as given in Hartshorne's "Deformation theory" proof of proposition $2.3$ $\endgroup$ – Naga Venkata Oct 22 '13 at 0:11

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