View source for Hilbert scheme

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A construction in algebraic geometry by means of which a set of closed subvarieties of a projective space with a given [[Hilbert polynomial|Hilbert polynomial]] can be endowed with the structure of an [[Algebraic variety|algebraic variety]]. More precisely, let $X$ be a projective scheme over a locally Noetherian scheme $S$ and let $\operatorname{Hilb}_{X/S}$ be the functor assigning to each $S$-scheme $S^*$ the set of closed subschemes $X^*=X\times_SS^*$ which are flat over $S^*$. The functor $\operatorname{Hilb}_{X/S}$ can be represented locally as a Noetherian scheme, known as the Hilbert scheme of $S$-schemes of $X$, and is denoted by $\operatorname{Hilb}(X/S)$ [[#References|[4]]]. By the definition of a [[Representable functor|representable functor]], for any $S$-scheme $S^*$ there is a bijection $\operatorname{Hilb}_{X/S}(S^*)=\Hom_S(S^*,\operatorname{Hilb}(X/S))$. In particular, if $S$ is the spectrum of a field $k$ (cf. [[Spectrum of a ring|Spectrum of a ring]]) and $X=P_k^n$ is a projective space over $k$, then the set of rational $k$-points of $\operatorname{Hilb}(P_k^n/k)$ is in one-to-one correspondence with the set of closed subvarieties in $P_k^n$.

For any polynomial $P\in\mathbf Q[x]$ with rational coefficients the functor $\operatorname{Hilb}_{X/S}$ contains a subfunctor $\operatorname{Hilb}_{X/S}^P$ which isolates in the set $\operatorname{Hilb}_{X/S}(S^*)$ the subset of subschemes $Z\subset X\times_SS^*$ such that for any point $s^*\in S^*$ the fibre $Z_{s^*}$ of the projection of $Z$ on $S^*$ has $P$ as its Hilbert polynomial. The functor $\operatorname{Hilb}_{S/X}^P$ can be represented by the Hilbert scheme $\operatorname{Hilb}^P(X/S)$, which is projective over $S$. The scheme $\operatorname{Hilb}(X/S)$ is the direct sum of the schemes $\operatorname{Hilb}^P(X/S)$ over all $P\in\mathbf Q(z)$. For any connected ground scheme $S$ the scheme $\operatorname{Hilb}^P(X/S)$ is also connected [[#References|[2]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) {{MR|0209285}} {{ZBL|0187.42701}} </TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> D. Mumford, "Geometric invariant theory" , Springer (1965) {{MR|0214602}} {{ZBL|0147.39304}} </TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top"> A. Grothendieck, "Techniques de construction et théorèmes d'existence en géométrie algébrique, IV : Les schémas de Hillbert" , ''Sém. Bourbaki'' , '''13''' : 221 (1960–1961) {{ZBL|0236.14003}}</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top"> R. Hartshorne, "Connectedness of the Hilbert scheme" ''Publ. Math. IHES'' , '''29''' (1966) pp. 5–48 {{MR|0213368}} {{ZBL|1092.14006}} {{ZBL|0994.14002}} </TD></TR>
<TR><TD valign="top">[5]</TD> <TD valign="top"> I.V. Dolgachev, "Abstract algebraic geometry" ''J. Soviet Math.'' , '''2''' : 3 (1974) pp. 264–303 ''Itogi Nauk. i Tekhn. Algebra Topol. Geom.'' , '''10''' (1972) pp. 47–112 {{MR|}} {{ZBL|1068.14059}} </TD></TR>
</table>