## A simple proof for the finiteness of GIT-quotients

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- by Alexander Schmitt PDF
- Proc. Amer. Math. Soc.
**131**(2003), 359-362 Request permission

## 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 Białynicki-Birula and Dolgachev and Hu.## References

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## Additional Information

**Alexander Schmitt**- Affiliation: Universität GH Essen, FB6 Mathematik & Informatik, D-45117 Essen, Germany
- MR Author ID: 360115
- ORCID: 0000-0002-4454-1461
- 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
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**131**(2003), 359-362 - MSC (1991): Primary 14L24, 14L30
- DOI: https://doi.org/10.1090/S0002-9939-02-06599-1
- MathSciNet review: 1933324