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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.

Question: What is the geometric quotient $Y_s / / G$ or $Y_{ss} / / G$?

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.

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  • $\begingroup$ Let $U$ denote the space of ternary cubics. Then from memory, $U^{ss}/SL_2$ is isomorphic to $\mathbf{P}^1$. I think a proof of this is probably in Dolgachev's Classical Invariant Theory. $\endgroup$
    – David Benjamin Lim
    Commented Jun 30, 2017 at 6:05

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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. See in particular Chapter 3, where the more general case of binary forms is discussed.

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