Homogeneous Hilbert scheme
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- by Amelia Álvarez, Fernando Sancho and Pedro Sancho PDF
- Proc. Amer. Math. Soc. 136 (2008), 781-790 Request permission
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{align*} \underline {\operatorname {HomHilb}}_{X/S} \colon \left [ \begin {array}{l} \text {Locally noetherian}\\ \text {$S$-schemes} \end{array} \right ] & \leadsto \text {Sets}\\ T & \leadsto \begin {array}{l} \text {closed subschemes of $X\times _ST$}\\ \text {flat and homogeneous over $T$} \end{array} \end{align*} 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.References
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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
- MR Author ID: 621464
- ORCID: 0000-0001-8915-2438
- Email: fsancho@usal.es
- Pedro Sancho
- Affiliation: Department of Mathematics, University of Extremadura, Avda Elvas s/n Badajoz, 06071 Spain
- Email: sancho@unex.es
- 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
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 781-790
- MSC (2000): Primary 14C05
- DOI: https://doi.org/10.1090/S0002-9939-07-09169-1
- MathSciNet review: 2361849