= First-order embedded deformation
A first-order embedded deformation of a <closed subscheme> $Z\subseteq X$ is a closed subscheme $Z_\varepsilon\subseteq X\times_k\operatorname{Spec}(k[\varepsilon]/(\varepsilon^2))$ flat over the <dual numbers> and with special fibre equal to the given embedded $Z$. It is a tangent vector of the <Hilbert scheme>, not a deformation modulo arbitrary automorphisms of the ambient scheme. These deformations are naturally the <global sections> of the <normal sheaf>.
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