PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 136, Number 3, March 2008, Pages 781–790 S 0002-9939(07)09169-1 Article electronically published on November 30, 2007 HOMOGENEOUS HILBERT SCHEME AMELIA ´ ALVAREZ, FERNANDO SANCHO, AND PEDRO SANCHO (Communicated by Michael Stillman) Abstract. Let S be a locally noetherian scheme and R an N-graded O S - algebra of finite type. We say that X = Spec R is a homogeneous variety over S. In this paper we prove that the functor HomHilb X/S :  Locally noetherian S-schemes  ❀ Sets T ❀ closed subschemes of X × S T flat and homogeneous over T is representable by an S-scheme that is a disjoint union of locally projective schemes over S. The proof is very simple, and it only makes use of the the- ory of graded modules and standard flatness criteria. From this, one obtains an elementary construction (which does not make use of cohomology) of the ordinary Hilbert scheme of a locally projective S-scheme. Introduction Let S be a locally noetherian scheme and R an N-graded O S -algebra of finite type. We say that X = Spec R is a homogeneous variety over S. In this paper we prove that the functor HomHilb X/S :  Locally noetherian S-schemes   Sets T  closed subschemes of X × S T that are flat and homogeneous over T is representable by an S-scheme that is a disjoint union of locally projective schemes over S (Theorem 1.9). This scheme is called the “Homogeneous Hilbert scheme of X”. The proof is very simple, and it only makes use of the theory of graded modules and standard flatness criteria. Let P(X) = Proj R. From the construction of the Homogeneous Hilbert scheme of X one obtains easily the construction of the Hilbert scheme of P(X). This construction is very simple and does not make use of cohomology. Received by the editors February 18, 2006 and, in revised form, October 6, 2006. 2000 Mathematics Subject Classification. Primary 14C05. Key words and phrases. Hilbert schemes. The second author was partially supported by the Spanish DGI research project BFM2003- 00097 and by JCYL research project SA114/04. c 2007 American Mathematical Society Reverts to public domain 28 years from publication 781 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use