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Let $G$ be a connected affine algebraic group over an algebraically closed field $K$ which acts algebraically on an affine $K$-variety $V:=\bigcup_{i=1}^{n} V_i$, where $V_i$'s are irreducible $G$-stable pairwise (disjoint) isomorphic as varieties. If one of the restriction $\pi_i:V_i\rightarrow V_i//G$ of the quotient morphism $\pi:V\rightarrow V//G$ is constant, does that imply any $\pi_j:V_j\rightarrow V_j//G, \forall j\ne i,$ is also constant? ($V//G$ is the GIT quotient of $V$).

It may be a trivial question, but not clear to me and I couldn't find out a counter example ( I believe its not true).

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  • $\begingroup$ I had asked this one day back, but deleted for the question was unclear. $\endgroup$
    – Poove
    Dec 10, 2011 at 19:15
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    $\begingroup$ I think in this case this is fine, but just as a general note: common practise is to edit the existing question (button below the tags). $\endgroup$
    – user9072
    Dec 10, 2011 at 19:19
  • $\begingroup$ The question is still unclear. Right now, I think you're asking "if one $G$-action on an affine variety has a dense orbit, then does any $G$-action on that variety?" which is obviously false (take the trivial action to be the other one). $\endgroup$
    – Ben Webster ♦
    Dec 10, 2011 at 20:50
  • $\begingroup$ @ Ben Webster: You are right, but here I have fixed a $G$ action on $V$ and so on each $V_i\subset V$, the restricted action of $G$. And $\pi_i:=\pi|V_i$. $\endgroup$
    – Poove
    Dec 11, 2011 at 8:37

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