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| In [[algebraic geometry]], a branch of [[mathematics]], a '''Hilbert scheme''' is a [[scheme theory|scheme]] that is the parameter space for the [[closed subscheme]]s of some projective space (or a more general projective scheme), refining the [[Chow variety]]. The Hilbert scheme is a disjoint union of [[projective subscheme]]s corresponding to [[Hilbert polynomial]]s. The basic theory of Hilbert schemes was developed by {{harvs|first=Alexander|last= Grothendieck|authorlink=Alexander Grothendieck|year=1961}}. [[Hironaka's example]] shows that non-projective varieties need not have Hilbert schemes.
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| ==Hilbert scheme of projective space==
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| The Hilbert scheme '''Hilb'''(''n'') of '''P'''<sup>''n''</sup> classifies closed subschemes of projective space in the following sense: For any [[locally Noetherian scheme]] ''S'', the set of ''S''-valued points
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| ::::Hom(''S'', '''Hilb'''(''n''))
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| of the Hilbert scheme is naturally isomorphic to the set of closed subschemes of '''P'''<sup>''n''</sup> × ''S'' that are [[flat morphism|flat]] over ''S''. The closed subschemes of '''P'''<sup>''n''</sup> × ''S'' that are flat over ''S'' can informally be thought of as the families of subschemes of projective space parameterized by ''S''. The Hilbert scheme '''Hilb'''(''n'') breaks up as a disjoint union of pieces '''Hilb'''(''n'', ''P'') corresponding to the Hilbert polynomial of the subschemes of projective space with Hilbert polynomial ''P''. Each of these pieces is projective over Spec('''Z''').
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| ===Construction===
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| Grothendieck constructed the Hilbert scheme '''Hilb'''(''n'')<sub>''S''</sub> of ''n''-dimensional projective space over a Noetherian scheme ''S'' as a subscheme of a [[Grassmannian]] defined by the vanishing of various [[determinant]]s. Its fundamental property is that for a scheme ''T'' over ''S'', it represents the functor whose ''T''-valued points are the closed subschemes of '''P'''<sup>''n''</sup> ×<sub>''S''</sub> ''T'' that are flat over ''T''.
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| If ''X'' is a subscheme of ''n''-dimensional projective space, then ''X'' corresponds to a graded ideal ''I''<sub>''X''</sub> of the polynomial ring ''S'' in ''n''+1 variables, with graded pieces ''I''<sub>''X''</sub>(''m''). For sufficiently large ''m'', depending only on the Hilbert polynomial ''P'' of ''X'', all higher cohomology groups of ''X'' with coefficients in ''O''(''m'') vanish, so in particular ''I''<sub>''X''</sub>(''m'') has dimension ''Q''(''m'') − ''P''(''m''), where ''Q'' is the Hilbert polynomial of projective space.
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| Pick a sufficiently large value of ''m''. The ''Q''(''m'') − ''P''(''m'')-dimensional space ''I''<sub>''X''</sub>(''m'') is a subspace of the ''Q''(''m'')-dimensional space ''S''(''m''), so represents a point of the Grassmannian ''G''(''Q''(''m'') − ''P''(''m''), ''Q''(''m'')). This will give an embedding of the piece of the Hilbert scheme corresponding to the Hilbert polynomial ''P'' into this Grassmannian.
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| It remains to describe the scheme structure on this image, in other words to describe enough elements for the ideal corresponding to it. Enough such elements are given by the conditions that the map from ''I''<sub>''X''</sub>(''m'') ⊗ ''S''(''k'') to ''S''(''k''+''m'') has rank at most dim(''I''<sub>''X''</sub>(''k''+''m'')) for all positive ''k'', which is equivalent to the vanishing of various determinants. (A more careful analysis shows that it is enough just to take ''k''=1.)
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| ===Variations===
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| The Hilbert scheme '''Hilb'''(''X'')<sub>''S''</sub> is defined and constructed for any projective scheme ''X'' in a similar way. Informally, its points correspond to closed subschemes of ''X''.
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| ===Properties===
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| {{harvtxt|Macaulay|1927}} determined for which polynomials the Hilbert scheme '''Hilb'''(''n'', ''P'') is non-empty, and {{harvtxt|Hartshorne|1966}} showed that if '''Hilb'''(''n'', ''P'') is non-empty then it is linearly connected. So two subschemes of projective space are in the same connected component of the Hilbert scheme if and only if they have the same Hilbert polynomial.
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| Hilbert schemes can have bad singularities, such as irreducible components that are non-reduced at all points.
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| They can also have irreducible components of unexpectedly high dimension. For example, one might expect the Hilbert scheme of ''d'' points (more precisely dimension 0, length ''d'' subschemes) of a scheme of dimension ''n'' to have dimension ''dn'', but if ''n''≥3 its irreducible components can have much larger dimension.
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| == Hilbert scheme of points on a manifold ==
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| "Hilbert scheme" sometimes
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| refers to the '''punctual Hilbert scheme''' of 0-dimensional subschemes on a scheme. Informally this can be thought of as something like finite collections of points on a scheme, though this picture can be very misleading when several points coincide.
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| There is a '''Hilbert-Chow morphism''' from the reduced Hilbert scheme of points to the Chow variety of cycles taking any 0-dimensional scheme to its associated 0-cycle. {{harvs|last=Fogarty|year1=1968|year2=1969|year3=1973}}.
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| The Hilbert scheme <math>M^{[n]}</math> of <math>n</math> points on <math>M</math>
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| is equipped with a natural morphism to an <math>n</math>-th
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| symmetric product of <math>M</math>. This morphism is
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| birational for ''M'' of dimension at most 2. For ''M'' of dimension at least 3 the morphism is not birational for large ''n'': the Hilbert scheme is in general reducible and has components of dimension much larger than that of the symmetric product.
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| The Hilbert scheme of points on a curve ''C'' (a dimension-1 complex manifold) is isomorphic to a [[Symmetric product of an algebraic curve|symmetric power]] of ''C''. It is smooth.
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| The Hilbert scheme of <math>n</math> points on a [[Complex surface|surface]] is
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| also smooth (Grothendieck). If <math>n=2</math>, it is obtained from <math>M\times M</math> by blowing up the diagonal and then
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| dividing by the <math>\mathbb{Z}_2</math> action induced by <math>(x,y) \mapsto (y,x)</math>. It was used by [[Mark Haiman]] in his proof
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| of the positivity of the coefficients of some [[Macdonald polynomial]]s.
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| The Hilbert scheme of a smooth manifold of dimension
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| 3 or more is usually not smooth.
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| == Hilbert schemes and hyperkähler geometry ==
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| Let <math>M</math> be a complex [[Kähler manifold|Kähler]] surface with <math>c_1=0</math> ([[K3 surface]] or a torus). The canonical bundle of <math>M</math> is trivial, as follows from [[Enriques-Kodaira classification|Kodaira classification of surfaces]]. Hence <math>M</math> admits a holomorphic [[Symplectic geometry|symplectic]] form. It was observed by [[Akira Fujiki|Fujiki]] (for <math>n=2</math>) and [[Arnaud Beauville|Beauville]] that <math>M^{[n]}</math> is also holomorphically symplectic. This is not very difficult to see, e.g., for <math>n=2</math>. Indeed, <math>M^{[2]}</math> is a blow-up of a symmetric square of <math>M</math>. Singularities of <math>Sym^2 M</math> are locally isomorphic to <math>\mathbf{C}^2 \times \mathbf{C}^2/\{\pm 1\}</math>. The blow-up of <math>\mathbf{C}^2/\{\pm 1\}</math> is <math>T^*\mathbf{P}^1(\mathbf{C})</math>, and this space is symplectic. This is used to show that the symplectic form is naturally extended to the smooth part of the exceptional divisors of <math>M^{[n]}</math>. It is extended to the rest of <math>M^{[n]}</math> by [[Hartogs' principle]].
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| A holomorphically symplectic, [[Kähler manifold]] is [[Hyperkähler manifold|hyperkähler]], as follows from [[Calabi conjecture|Calabi-Yau theorem]]. Hilbert schemes of points on [[K3 surface|K3]] and a 4-dimensional torus give two series of examples of [[hyperkähler manifold]]s: a Hilbert scheme of points on K3 and a generalized Kummer manifold.
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| ==References==
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| *{{Citation | last1=Beauville | first1=Arnaud | title=Variétés Kähleriennes dont la première classe de Chern est nulle | mr=730926 | year=1983 | journal=Journal of Differential Geometry | volume=18 | issue=4 | pages=755–782}}
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| *{{springer|id=H/h047320|authorlink=Dolgachev|author=I. Dolgachev|title=Hilbert scheme}}
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| *{{Citation | last1=Fantechi | first1=Barbara | last2=Göttsche | first2=Lothar | last3=Illusie | first3=Luc | author3-link=Luc Illusie | last4=Kleiman | first4=Steven L. | author4-link=Steven Kleiman | last5=Nitsure | first5=Nitin | last6=Vistoli | first6=Angelo | title=Fundamental algebraic geometry | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-3541-8 | mr=2222646 | year=2005 | volume=123|url=http://books.google.com/?id=JhDloxGpOA0C}}
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| *{{Citation | last1=Fogarty | first1=John | title=Algebraic families on an algebraic surface | mr=0237496 | year=1968 | journal=[[American Journal of Mathematics]] | volume=90 | pages=511–521 | doi=10.2307/2373541 | jstor=2373541 | issue=2 | publisher=The Johns Hopkins University Press}}
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| *{{Citation | last1=Fogarty | first1=John | title=Truncated Hilbert functors | url=http://gdz.sub.uni-goettingen.de/no_cache/en/dms/load/img/?IDDOC=252601 | mr=0244268 | year=1969 | journal=[[Journal für die reine und angewandte Mathematik]] | volume=234 | pages=65–88}}
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| *{{Citation | last1=Fogarty | first1=John | title=Algebraic families on an algebraic surface. II. The Picard scheme of the punctual Hilbert scheme | mr=0335512 | year=1973 | journal=[[American Journal of Mathematics]] | volume=95 | pages=660–687 | doi=10.2307/2373734 | jstor=2373734 | issue=3 | publisher=The Johns Hopkins University Press}}
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| *{{Citation | last1=Göttsche | first1=Lothar | title=Hilbert schemes of zero-dimensional subschemes of smooth varieties | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-57814-7 | doi=10.1007/BFb0073491 | mr=1312161 | year=1994 | volume=1572}}
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| *{{Citation | last1=Grothendieck | first1=Alexander | author1-link=Alexander Grothendieck | series=Séminaire Bourbaki 221 | year=1961 | title=Techniques de construction et théorèmes d'existence en géométrie algébrique. IV. Les schémas de Hilbert|url=http://www.numdam.org/item?id=SB_1960-1961__6__249_0 }} Reprinted in {{Citation | title=Séminaire Bourbaki, Vol. 6 | publisher=[[Société Mathématique de France]] | location=Paris | mr=1611822 | year=1995 | pages=249–276|isbn= 2-85629-039-6 | author=Adrien Douady, Roger Godement, Alain Guichardet ... }}
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| *{{Citation | last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=Connectedness of the Hilbert scheme | url=http://www.numdam.org/item?id=PMIHES_1966__29__5_0 | mr=0213368 | year=1966 | journal=[[Publications Mathématiques de l'IHÉS]] | issue=29 | pages=5–48}}
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| *{{citation|authorlink=F. S. Macaulay|last=Macaulay|first= F. S.|year=1927
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| |title=Some properties of enumeration in the theory of modular systems
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| |journal=Proceedings L. M. S. Series 2 |volume=26|pages= 531–555|doi=10.1112/plms/s2-26.1.531}}
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| *{{Citation | last1=Mumford | first1=David | author1-link=David Mumford | title=Lectures on Curves on an Algebraic Surface | publisher=[[Princeton University Press]] | series=Annals of Mathematics Studies | isbn=978-0-691-07993-6 | volume=59}}
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| *{{Citation | last1=Nakajima | first1=Hiraku | title=Lectures on Hilbert schemes of points on surfaces | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=University Lecture Series | isbn=978-0-8218-1956-2 | mr=1711344 | year=1999 | volume=18}}
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| *{{Citation | last1=Nitsure | first1=Nitin | title=Fundamental algebraic geometry | arxiv=math/0504590 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Math. Surveys Monogr. | mr=2223407 | year=2005 | volume=123 | chapter=Construction of Hilbert and Quot schemes | pages=105–137}}
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| ==External links==
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| *{{citation|last=Bertram|first=Aaron|url=http://www.math.utah.edu/~bertram/courses/hilbert/|title=Construction of the Hilbert scheme|year=1999|accessdate=2008-09-06}}
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| [[Category:Scheme theory]]
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| [[Category:Algebraic geometry]]
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| [[Category:Differential geometry]]
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| [[Category:Moduli theory]]
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Nice to meet you, my name is Refugia. I am a meter reader but I plan on altering it. To play baseball is the pastime he will by no means stop doing. North Dakota is her birth location but she will have to transfer one day or another.
my weblog; std home test (similar internet site)