I'm trying to understand the proof of a theorem by Van Den Bergh, which is Proposition 6 in the paper Bessenrodt, Christine and Lieven Le Bruyn. “Stable rationality of certain PGLn-quotients.” Inventiones mathematicae 104 (1991). It's stated as follows:
Theorem:
Assume that a reductive group $G$ acts on a factorial affine variety $X$ with finite generic stabilizer. Then there exists an affine $G$-invariant open subvariety $V$ of $X$ such that generic orbits are closed in $V$.
Proof:
We may assume that $G$ is connected. Let $T = Z(G)^0$ and $G_0 = [G, G]$, then $G_0$ is semisimple, $T$ is a torus, $G = TG_0$ and $T\cap G_0$ is finite. Let $\phi: X \rightarrow X/G_0$ be the quotient map. $T$ acts on $X/G_0$ and hence, there exists a $T$-invariant open subset $U$ of $X/G_0$ such that T-orbits are closed. But, then according to [15, w Th. 5] there exists a T-invariant affine open $U'$ in $U$. Let $V=\phi^{-1}(U')$, then this is an affine $G$-invariant open subset of $X$ as $\phi$ is affine. Now take $x\in V$ a generic point. Then using [18, app. FI, we have that $Gx = TG_0x = T\phi^{-1}(\phi(x)) = \phi^{-1}(T\phi(x))$ is the inverse image of a closed set, hence it is closed.
My question:
- Can someone explain me the part written in bold? Why does there exist such an open set $U$?
- Also, can we give some description of the open set $V=\phi^{-1}(U')$? Maybe something like-- $V$ is given by the non-vanishing of a $G$-invariant function or $G_0$-invariant function.
