Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Homogeneous Hilbert scheme


Authors: Amelia Álvarez, Fernando Sancho and Pedro Sancho
Journal: Proc. Amer. Math. Soc. 136 (2008), 781-790
MSC (2000): Primary 14C05
Published electronically: November 30, 2007
MathSciNet review: 2361849
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Abstract: Let $ S$ be a locally noetherian scheme and $ R$ an $ \mathbb{N}$-graded $ \mathcal O_S$-algebra of finite type. We say that $ X=\operatorname{Spec}R$ is a homogeneous variety over $ S$. In this paper we prove that the functor

\begin{displaymath}\begin{array}{cll} \underline{\operatorname{HomHilb}}_{X/S} \... ...es_ST$}\\ & & \text{flat and homogeneous over $T$} \end{array}\end{displaymath}

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 theory 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.


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Additional Information

Amelia Álvarez
Affiliation: Department of Mathematics, University of Extremadura, Avda Elvas s/n, Badajoz, 06071 Spain
Email: aalarma@unex.es

Fernando Sancho
Affiliation: Department of Mathematics, University of Salamanca, Plaza de la Merced 1-4, 37008 Salamanca, Spain
Email: fsancho@usal.es

Pedro Sancho
Affiliation: Department of Mathematics, University of Extremadura, Avda Elvas s/n Badajoz, 06071 Spain
Email: sancho@unex.es

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09169-1
Keywords: Hilbert schemes.
Received by editor(s): February 18, 2006
Received by editor(s) in revised form: October 6, 2006
Published electronically: November 30, 2007
Additional Notes: The second author was partially supported by the Spanish DGI research project BFM2003-00097 and by JCYL research project SA114/04.
Communicated by: Michael Stillman
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.