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A simple proof for the finiteness of GIT-quotients

Authors:	Alexander Schmitt

Institution:	Universität GH Essen, FB6 Mathematik & Informatik, D-45117 Essen, Germany

Abstract:	Let $G\times X\longrightarrow X$ be an action of the reductive group on the projective scheme . 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 $\l$ ynicki-Birula and Dolgachev and Hu.

Keywords:	Linearization semistable points torus action Hilbert-Mumford criterion

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