TITLE: "Interesting" projective varieties being quotients of $\mathbb{A}^n\setminus \{0\}$ by an action of an algebraic group?
QUESTION [6 upvotes]: The algebraic (multiplicative) group $G^m$ acts on $\mathbb{A}^n$ (diagonally) and the quotient of $\mathbb{A}^n\setminus \{0\}$ by $G_m$ is $\mathbb{P}^{n-1}$ (which is a proper variety). I would like to find a "more interesting" example of an action of an  algebraic groups $G$ on  $\mathbb{A}^n$ such that the quotient of $\mathbb{A}^n\setminus A$ by $G$ is proper, where $A$ is a certain "small" subvariety of $\mathbb{A}^n$ (say, over the field of complex numbers). Under these conditions does the quotient have to be toric (and does $G$ have to be a torus)?
I would be deeply grateful for any hints or references (and I know very little on algebraic groups and toric varieties)!

REPLY [4 votes]: As Daniel Loughran pointed out, this situation has been largely investigated (and generalized) in recent years. I don't know if the theory of Cox Rings is overkill for your situation, but let me refer you to this book that is treating the subject systematically and not assuming much prior knowledge. The first chapter is available here.
(I would've written this as a comment as it's not really an answer to your question, but apparently you have to have asked or answered a few questions before being able to do that.)