A simple proof for the finiteness of GIT-quotients

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.