Moduli Spaces of Unordered $n\ge5$ Points on the Riemann Sphere and Their Singularities

TL;DR

This paper characterizes the stabilizer groups of finite point configurations on the Riemann sphere, describes their relation to the moduli space singularities, and explicitly analyzes these structures for cases with 5 and 6 points.

Contribution

It provides necessary and sufficient conditions for stabilizers to be conjugate to finite subgroups of PSL(2,C), and explicitly describes singularities and stabilizer representations for n=5 and 6.

Findings

Characterization of stabilizer groups for finite point sets on the sphere.

Conditions for stabilizers to be conjugate to finite subgroups of PSL(2,C).

Explicit analysis of singularities and stabilizer actions for n=5 and 6.

Abstract

For $n\ge5$, it is well known that the moduli space $\mathfrak{M_{0,\:n}}$ of unordered $n$ points on the Riemann sphere is a quotient space of the Zariski open set $K_n$ of $\mathbb C^{n-3}$ by an $S_n$ action. The stabilizers of this $S_n$ action at certain points of this Zariski open set $K_n$ correspond to the groups fixing the sets of $n$ points on the Riemann sphere. Let $\alpha$ be a subset of $n$ distinct points on the Riemann sphere. We call the group of all linear fractional transformations leaving $\alpha$ invariant the stabilizer of $\alpha$, which is finite by observation. For each non-trivial finite subgroup $G$ of the group ${\rm PSL}(2,{\Bbb C})$ of linear fractional transformations, we give the necessary and sufficient condition for finite subsets of the Riemann sphere under which the stabilizers of them are conjugate to $G$. We also prove that there does exist some finite subset of the Riemann sphere whose stabilizer coincides with $G$. Next we obtain the irreducible decompositions of the representations of the stabilizers on the tangent spaces at the singularities of $\mathfrak{M_{0,\:n}}$. At last, on $\mathfrak{M_{0,\:5}}$ and $\mathfrak{M_{0,\:6}}$, we work out explicitly the singularities and the representations of their stabilizers on the tangent spaces at them.