# “Interesting” projective varieties being quotients of $\mathbb{A}^n\setminus \{0\}$ by an action of an algebraic group?

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)!

• 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. – Allen Knutson Oct 7 '15 at 9:17
• Thank you!! Funnily enough, just before reading your comment I was wondering whether $A$ is small enough in this example.:) – Mikhail Bondarko Oct 7 '15 at 9:36
• Does "small" for you just mean that the quotient is proper? – Allen Knutson Oct 7 '15 at 10:28
• No, "small" means "of small dimension" (and I am thinking what does the latter "small" means). – Mikhail Bondarko Oct 7 '15 at 10:55
• – Lucas Kaufmann Oct 7 '15 at 13:35

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:
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.