Hilbert scheme

A construction in algebraic geometry by means of which a set of closed subvarieties of a projective space with a given Hilbert polynomial can be endowed with the structure of an 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 $\operatorname{Hilb}(X/S)$ [4]. By the definition of a 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) 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 [2].

References

[1]	D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) MR0209285 Zbl 0187.42701
[2]	D. Mumford, "Geometric invariant theory" , Springer (1965) MR0214602 Zbl 0147.39304
[3]	A. Grothendieck, "Techniques de construction et théorèmes d'existence en géométrie algébrique, IV: Les schémas de Hillbert" , Sem. Bourbaki , 13 : 221 (1960–1961)
[4]	R. Hartshorne, "Connectedness of the Hilbert scheme" Publ. Math. IHES , 29 (1966) pp. 5–48 MR0213368 Zbl 1092.14006 Zbl 0994.14002
[5]	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 Zbl 1068.14059