Hilbert functor (${\mathrm{Hilb}}_{X/k}$)

For a projective scheme $X$ over a field $k$, the Hilbert functor assigns to a $k$-scheme $T$ the set of closed subschemes $W\subseteq X{\times }_{k}T$ which are of finite presentation, proper and flat over $T$. Pullback of families makes it a contravariant functor. After choosing a very ample line bundle, requiring a fixed fibre Hilbert polynomial $P$ gives the subfunctor represented by ${\mathrm{Hilb}}^P(X)$. The unrestricted functor is represented by the disjoint union of these Hilbert schemes.

A first-order embedded deformation of a closed subscheme $Z\subseteq X$ is a closed subscheme $Z_{\epsilon }\subseteq X{\times }_k\mathrm{Spec}(k[\epsilon ]/({\epsilon }^2))$ 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.

For a closed subscheme $Z\subseteq X$ with ideal sheaf $\mathcal{I}$, one has $T_{[Z]}\mathrm{Hilb}(X)\cong {\mathrm{Hom}}_{{\mathcal{O}}_X}(\mathcal{I},i_{\ast }{\mathcal{O}}_Z)\cong H^0(Z,{\mathcal{N}}_{Z/X})$. Locally a flat deformation ideal $J\subseteq A[\epsilon ]/({\epsilon }^2)$ satisfies $J\cap \epsilon A=\epsilon I$. Lifting $f\in I$ to $f+\epsilon g\in J$ gives a well-defined map $f\mapsto g\mathrm{mod}I$ killing $I^2$. Conversely the graph condition $g\mathrm{mod}I=\varphi (f)$ defines an ideal with flat quotient by the flatness criterion over dual numbers. These operations commute with restriction and are inverse, giving the global identification without a smoothness assumption.