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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)!

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    $\begingroup$ Anything on "geometric invariant theory", e.g. the book with that title, will have lots more examples, where $A$ is the "unstable set". If $G$ contains dilation then the GIT quotient will be proper. For a non-torus example, let $GL(k)$ act on $k\times n$ matrices, and let $A$ be the set of matrices of rank $< k$. Then the quotient is $Gr(k,n)$. Of course, maybe this $A$ isn't "small" enough for you. $\endgroup$ – Allen Knutson Oct 7 '15 at 9:17
  • $\begingroup$ Thank you!! Funnily enough, just before reading your comment I was wondering whether $A$ is small enough in this example.:) $\endgroup$ – Mikhail Bondarko Oct 7 '15 at 9:36
  • $\begingroup$ Does "small" for you just mean that the quotient is proper? $\endgroup$ – Allen Knutson Oct 7 '15 at 10:28
  • $\begingroup$ No, "small" means "of small dimension" (and I am thinking what does the latter "small" means). $\endgroup$ – Mikhail Bondarko Oct 7 '15 at 10:55
  • $\begingroup$ related mathoverflow.net/questions/30907/… $\endgroup$ – Lucas Kaufmann Oct 7 '15 at 13:35
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By the way your question is phrased, it seems that you might be familiar with the following construction. But anyway, any toric variety can be realised as the quotient of $\mathbb{A}^n \setminus A$ by the action of an algebraic torus (for some $n$ and $A$). This is proved in the famous paper of Cox:

The Homogeneous Coordinate Ring of a Toric Variety.

This theory has been generalised much by the theory of Cox rings and universal torsors. This says that any "suitably nice" variety $X$ (namely, a Mori dream space) is a quotient of $\mathrm{Spec}(\mathrm{Cox}(X)) \setminus A$ (for some $A$) by the action of the Néron-Severi torus of $X$. Moreover, $\mathrm{Spec}(\mathrm{Cox}(X))$ is isomorphic to an affine space if and only if $X$ is a toric variety.

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  • $\begingroup$ Thank you!! I really know very little on this subject; so any references and key words may help. $\endgroup$ – Mikhail Bondarko Oct 7 '15 at 9:31
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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.)

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