View source for 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|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" />.

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]]].

====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>