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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h0473201.png" /> be a projective scheme over a locally Noetherian scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h0473202.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h0473203.png" /> be the functor assigning to each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h0473204.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h0473205.png" /> the set of closed subschemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h0473206.png" /> which are flat over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h0473207.png" />. The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h0473208.png" /> can be represented locally as a Noetherian scheme, known as the Hilbert scheme of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h0473209.png" />-schemes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732010.png" />, and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732011.png" /> [[#References|[4]]]. By the definition of a [[Representable functor|representable functor]], for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732012.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732013.png" /> there is a bijection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732014.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732015.png" /> is the spectrum of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732016.png" /> (cf. [[Spectrum of a ring|Spectrum of a ring]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732017.png" /> is a projective space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732018.png" />, then the set of rational <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732019.png" />-points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732020.png" /> is in one-to-one correspondence with the set of closed subvarieties in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732021.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732022.png" /> with rational coefficients the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732023.png" /> contains a subfunctor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732024.png" /> which isolates in the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732025.png" /> the subset of subschemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732026.png" /> such that for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732027.png" /> the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732028.png" /> of the projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732029.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732030.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732031.png" /> as its Hilbert polynomial. The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732032.png" /> can be represented by the Hilbert scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732033.png" />, which is projective over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732034.png" />. The scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732035.png" /> is the direct sum of the schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732036.png" /> over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732037.png" />. For any connected ground scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732038.png" /> the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047320/h04732039.png" /> is also connected [[#References|[2]]].
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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====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> D. Mumford,   "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> D. Mumford,   "Geometric invariant theory" , Springer (1965)</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" , ''Sem. Bourbaki'' , '''13''' : 221 (1960–1961)</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</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</TD></TR></table>
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<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>
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<TR><TD valign="top">[2]</TD> <TD valign="top"> D. Mumford, "Geometric invariant theory" , Springer (1965) {{MR|0214602}} {{ZBL|0147.39304}} </TD></TR>
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<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>
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<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>
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<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>
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</table>

Latest revision as of 05:48, 17 April 2024

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" , Sém. Bourbaki , 13 : 221 (1960–1961) Zbl 0236.14003
[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
How to Cite This Entry:
Hilbert scheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hilbert_scheme&oldid=13116
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article