Three theorems on existence of good quotients (Q1359523)

Let a reductive group \(G\) act on an algebraic variety \(X\). A \(G\)-morphism \(\pi:X\to Y\), where \(Y\) is an algebraic \(G\)-space with the trivial action of \(G\), is said to be a good quotient of \(G\), if \(\pi\) is affine and \(\pi_*({\mathcal O}_X)^G={\mathcal O}_Y\). The first of the three theorems mentioned in the title, says that if, for every curve \(C\) embedded in \(X\), there exists a good quotient of \(\overline{G}\overline{C}\) by \(G\) then there exists a good quotient of \(X\) by \(G\). -- The second theorem is, that if \(X\) is normal and there exists an affine \(G\)-morphism of \(X\) onto an algebraic space \(Y\) with the trivial action of \(G\), then there exists a good quotient of \(X\) by \(G\). -- The third theorem says that, if for any pair of points \((x,y)\) in \(X\), there exists their common \(G\)-invariant neighborhood \(U_{x,y}\) with a good quotient by \(G\), then there exists a good quotient of \(X\) by \(G\). Though the theorems have different character their proofs are related to each other.