We study the invariants of an algebraic group action on an affine variety via separating morphisms, that is, dominant G-invariant morphism to another affine variety such that points which are separated by some invariant have distinct image. This is a more geometric take on the study of separating invariants, a new trend in invariant theory initiated by Derksen and Kemper.
In this talk, I will discuss some results which indicate that the fact that the invariants are not always finitely generated is less significant than the fact that what we would want to call the quotient morphism is not always surjective.
(Joint work with Hanspeter Kraft)