Orbits and invariants of visible group actions
Let \(X\) be an irreducible affine algebraic variety endowed with a regular action of an affine algebraic group \(G\). The author calls such an action visible if the field of \(G\)-invariant rational functions on \(X\) is the field of fractions of the algebra of \(G\)-invariant regular functions on \(X\). It is shown that if the action is not visible, then there is a visible action of \(G\) on a variety \(Z\) and a \(G\)-equivariant birational morphism \(Z\to X\). This is used for obtaining visible \(G\)-invariant open subsets of \(X\). Visibility is also explored in the presence of the following properties: (1) stability of the actions of \(G\) on \(X\), (2) the existence of a regular semi-invariant \(f\) such that the \(G\)-action on \(X_f\) is visible; (3) reductivity of \(G\).
scientific article; zbMATH DE number 6136957
scientific article