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).

editthe existing question (button below the tags). – quid Dec 10 '11 at 19:19