I want to study the space $Y$ of all un-ordered $d$-tuples of points on $\mathbb P^1$. By considering the space $V\_d$ of homogeneous polynomials of degree $d$ in two variables, one may identify $Y$ with $\mathbb P(V\_d)\equiv \mathbb P^d$. There is a natural $SL\_2$ action on $\mathbb P(V\_d)\equiv Y$ via change of coordinates. For simplicity, the field is assumed to be $\mathbb C$, and let me assume $d=4$ in the sequel.


I want to study the subspace $Y\_s \subset Y$ consisting of un-ordered 4-tuples of pairwise distinct points. Indeed $Y\_s$ is the locus of stable points according to Hilbert-Mumford criterion. Put $G=SL\_2$. Denote $Y\_{ss}$ the locus of semi-stable points.



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> **Question:** What is the geometric quotient $Y\_s / / G$ or $Y\_{ss} / / G$?
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I guess the first answer should be $\mathbb A^1$. Intuitively, $Y\_s / /G$ characterizes the orbits. If I specify three points of the tuple to be $0,1,\infty$, then the fourth point can vary within almost all of $\mathbb A^1$. The issue might be that here the tuples are un-ordered. And the second answer, I guess, might be $\mathbb P^1$, since $Y\_{ss} / /G $ is a projective variety by its construction. (I am a beginner so I cannot guarantee what I said was completely correct.) Thank you for your time.



un-ordered distinct $n$-tuples of points on $\mathbb P^1$
The quotient is actually isomorphic to $\mathbb{P}^1$. 


You can find a proof in the notes by G. Ottaviani and J. Vallès [*Moduli of vector bundles and group actions*](https://www.google.it/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0ahUKEwiU86yRjuXUAhWFWRQKHTbBDysQFggiMAA&url=http%3A%2F%2Fweb.math.unifi.it%2Fusers%2Fottavian%2Fwykno%2Fwykno24feb.pdf&usg=AFQjCNEOdC7RMDer6In2Xv-3VK0CJjq_cw&cad=rja). See in particular Chapter 3, where the more general case of binary forms is discussed.


