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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A simple proof for the finiteness of GIT-quotients


Author: Alexander Schmitt
Journal: Proc. Amer. Math. Soc. 131 (2003), 359-362
MSC (1991): Primary 14L24, 14L30
Published electronically: June 3, 2002
MathSciNet review: 1933324
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Abstract: Let $G\times X\longrightarrow X$ be an action of the reductive group $G$ on the projective scheme $X$. For every linearization $\sigma$ of this action in an ample line bundle, there is an open set $X_\sigma^{\mathrm{ss}}$ of $\sigma$-semistable points. We provide an elementary and geometric proof for the fact that there exist only finitely many open sets of the form $X_\sigma^{\mathrm{ss}}$. This observation was originally due to Bia\lynicki-Birula and Dolgachev and Hu.


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Additional Information

Alexander Schmitt
Affiliation: Universität GH Essen, FB6 Mathematik & Informatik, D-45117 Essen, Germany

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06599-1
PII: S 0002-9939(02)06599-1
Keywords: Linearization, semistable points, torus action, Hilbert-Mumford criterion
Received by editor(s): April 17, 2001
Received by editor(s) in revised form: September 17, 2001
Published electronically: June 3, 2002
Communicated by: Michael Stillman
Article copyright: © Copyright 2002 American Mathematical Society