Group actions, divisors, and plane curves

Abstract

After a general discussion of group actions, orbifolds, and weak orbifolds, this note will provide elementary introductions to two basic moduli spaces over the real or complex numbers: first the moduli space of effective divisors with finite stabilizer on the projective space $\mathbb{P}^1$, modulo the group of projective transformations of $\mathbb{P}^1$; and then the moduli space of curves (or more generally effective algebraic $1$-cycles) with finite stabilizer in $\mathbb{P}^2$, modulo the group of projective transformations of $\mathbb{P}^2$. It also discusses automorphisms of curves and the topological classification of smooth real curves in $\mathbb{P}^2$.

1. Introduction

A basic objective of this paper is to provide an elementary introduction to the moduli space of curves of degree $n$ in the real or complex projective plane, modulo the action of the group of projective transformations. However, in order to prepare for this, we first give an exposition of the general theory of smooth group actions and orbifolds. As a further introduction, we describe the theory of effective divisors on the real or complex projective line modulo projective transformations. We also discuss a number of related topics, including automorphism groups of curves and the topology of smooth curves in the real projective plane. The different sections are largely independent of each other.

§2. Group actions and orbifolds

We consider smooth actions of Lie groups on smooth Hausdorff manifolds. Since these actions are not always proper, we introduce the concepts of locally proper action, and weakly locally proper action. We introduce the usual concept of orbifold,⁠¹ as well the concept of weak orbifold for quotient spaces which have only some of the standard orbifold properties, and we prove the following:

¹

Most authors require orbifolds to be Hausdorff spaces, however we will allow orbifolds which are only locally Hausdorff.

✖

For a proper/locally proper/or weakly proper action the quotient space is a Hausdorff orbifold/locally Hausdorff orbifold/or locally Hausdorff weak orbifold, respectively.

§3. Divisors on $\mathbb{P}^1$ and the moduli space $\mathfrak{M}_n$.

Working either over the real or complex numbers, the space $\mathfrak{M}_n$ of effective divisors of degree $n$ with finite stabilizer on the projective line $\mathbb{P}^1$, modulo projective transformations, is a $\mathrm{T}_1$ space for every $n$, but it is a Hausdorff orbifold only for $n \leq 4$. In the complex case the space $\mathfrak{M}_4(\mathbb{C})$ is homeomorphic to $\mathbb{P}^1(\mathbb{C})$, but in the real case $\mathfrak{M}_4(\mathbb{R})$ is homeomorphic to a closed line segment in $\mathbb{P}^1(\mathbb{R})$. In both the real and complex cases, there is just one point of $\mathfrak{M}_4$ which is “improper” in the sense that the action of the group of projective transformation at corresponding divisors is not proper. On the other hand, for $n>4$, there is a unique maximal open subset $\mathfrak{M}_n^\mathsf{Haus}\subset \mathfrak{M}_n$ which is a Hausdorff orbifold, but $\mathfrak{M}_n$ is not even locally Hausdorff at points outside of $\mathfrak{M}_n^\mathsf{Haus}$. This subset $\mathfrak{M}_n^\mathsf{Haus}$ is compact for $n$ odd, but not for $n$ even.

The proofs make use of a Distortion Lemma for automorphisms of $\mathbb{P}^1$ which describes the way in which an automorphism that lies outside a large compact subset of $\operatorname {PGL}_2$ must distort the geometry. The proofs also make use of two familiar projective invariants associated with a 4-tuple of points in ${\mathbb{P}}^1$. The cross-ratio depends on the ordering of the four points, while the shape invariant $\mathbf{J}=\mathbf{J}(x, y, z, w)$ (the classical $j$-invariant times a convenient constant) is invariant under permutations of the four points.

In the complex case, a close relative of our $\mathfrak{M}_n(\mathbb{C})$ is the classical moduli space ${\mathcal{M}}_{0, n}$ consisting of closed Riemann surfaces of genus zero provided with an ordered list of $n\geq 3$ distinct points, where two such marked Riemann surfaces are identified if there is a conformal isomorphism taking one to the other. We also discuss the compactification $\overline{\mathcal{M}}_{0,n}(\mathbb{C})$, a beautiful object introduced by Knudsen, based on ideas of Grothendieck, Deligne, and Mumford. (See also Keel Ke and Etingof, Henriques, Kamnitzer, and Rains EHKR for the analogous space $\overline{\mathcal{M}}_{0,n}(\mathbb{R})$.) In both the real and complex cases, there are $\displaystyle {n\choose p}$ associated embeddings

$$\begin{equation*} \overline{\mathcal{M}}_{0,\, p+1} \times \overline{\mathcal{M}}_{0,\, q+1} \hookrightarrow \overline{\mathcal{M}}_{0,\,n}, \end{equation*}$$

where $p+q=n$. As an example, in the real case for $p=2$ and $q=3$, there are ten essentially different embeddings of the circle $\overline{\mathcal{M}}_{0,\,3}\times \overline{\mathcal{M}}_{0,\,4}$ into the surface $\overline{\mathcal{M}}_{0,\,5}$, cutting it up into 12 hyperbolic pentagons, and thus presenting it as a hyperbolic analogue of the dodecahedron. The space $\overline{\mathcal{M}}_{0,\,6}({\mathbb{R}})$ is a 3-manifold which has a Jaco-Shalen-Johanssen decomposition, showing that it has some combination of hyperbolic and flat geometry. More explicitly, if we remove the ten tori corresponding to the embeddings of $\overline{\mathcal{M}}_{0,\,4}\times \overline{\mathcal{M}}_{0,\,4}$ into ${\overline{\mathcal{M}}}_{0,\,6}$, then the remainder can be given the structure of a complete hyperbolic 3-manifold with 20 infinite cusps.

§4. Curves (or $1$-cycles) in $\mathbb{P}^2$ and their moduli space

The space of all curves of degree $n$ in $\mathbb{P}^2$ can be conveniently considered as a dense open subset of a projective space of dimension $n(n+3)/2$ whose elements are formal linear combinations of curves in $\mathbb{P}^2$. These are called effective (algebraic) $1$-cycles. For $n\ge 3$, we study the moduli space ${\mathbb{M}}_n$ consisting of all projective equivalence classes of effective $1$-cycles ${\mathcal{C}}$ of degree $n$ with finite stabilizer. (For $n<3$, there are no curves with finite stabilizer.) This moduli space has been studied by many authors. (See for example Mumford’s discussion of hypersurfaces Mu, p. 79, which includes not only curves in $\mathbb{P}^2$ but also divisors in $\mathbb{P}^1$.) However, our presentation is more elementary, and is addressed to nonspecialists.

§5. Cubic curves

The space ${\mathbb{M}}_3({\mathbb{C}})$ is homeomorphic to the topological 2-sphere consisting of all ratios $(a^3:b^2) \in {\mathbb{P}}^1({\mathbb{C}})$, corresponding to curves $y^2=x^3+ax+b$. It is isomorphic as an orbifold to the moduli space ${\mathfrak{M}}_4({\mathbb{C}})$ for divisors of degree four on $\mathbb{P}^1(\mathbb{C})$. In the real case, the space $\mathbb{M}_3(\mathbb{R})$ is diffeomorphic to the unit circle. We describe the resulting circle of real cubic curves in several different ways, not only in terms of the normal form $y^2=x^3+ax+b$ with $a^2+b^2=1$, but also in terms of the Hesse normal form $x^3+y^3+z^3=3kxyz$, with $k\in \mathbb{R}\cup \{\infty \}$, and in terms of a convenient flex-slope normal form, parametrized by the slope $s$ at a flex point.

§6. Degree at least four.

When $n\geq 5$, we prove that the moduli space ${\mathbb{M}}_n({\mathbb{C}})$ is not a Hausdorff space. On the other hand, we provide two different procedures for describing large open subsets of ${\mathbb{M}}_n({\mathbb{C}})$ which are Hausdorff orbifolds. One of them is based on studying the distribution of virtual flex points, that is points which are either flex points or which yield flex points under perturbation. The other will be discussed in §7.

The open subset ${\mathbb{M}}_n^\mathsf{sm}\subset \mathbb{M}_n(\mathbb{C})$ corresponding to smooth curves in $\mathbb{P}^2(\mathbb{C})$ is a Hausdorff space, which maps naturally into the classical moduli space $\mathcal{M}_{\mathfrak{g}(n)}$ consisting of all isomorphism classes of compact Riemann surfaces of genus $\mathfrak{g}(n)={n-1\choose 2}$. This map is injective for all $n\geq 3$ (compare Chang Ch), but is an isomorphism only for $n=3$.

§7. Singularity genus and proper action

This section provides a different proof that large subsets of $\mathbb{M}_n(\mathbb{C})$ are Hausdorff orbifolds, based on the genus invariant for singular points. Both proofs make use of a Distortion Lemma for automorphisms of $\mathbb{P}^2$ which lie outside a large compact set of group elements. These arguments apply only to points of moduli space which correspond to (possibly singular) curves, but there is a brief discussion of extending the proof to more general $1$-cycles.

§8. Infinite automorphism groups.

Although we are primarily concerned with curves which have finite stabilizer, this section studies the opposite case of curves with infinitely many automorphisms. Following Klein and Lie, we call these $\mathrm{W}$-curves. We provide an explicit catalog of all such curves. (Compare Aluffi and Faber AF1.)

§9. Finite automorphism groups.

The study of finite automorphism groups of curves is closely related to the study of the moduli space $\mathbb{M}_n$, since any curve with extra automorphisms gives rise to a singular point in moduli space, or at least to a ramified point. For each degree $d$ and prime $p$, we provide an explicit criterion for deciding whether there exists a smooth curve of degree $d$ with a projective automorphism of period $p$, and we compare this with a corresponding statement for automorphisms of arbitrary compact Riemann surfaces. We show that an arbitrary finite subgroup of $\operatorname {PGL}_3(\mathbb{C})$ is the full projective automorphism group for some smooth curve in $\mathbb{P}^2(\mathbb{C})$, but show (following Chang Ch) that a generic⁠² curve of degree four has no nontrivial projective automorphism.

²

We say that a statement is true for a generic curve if it is true for all curves in some set which is dense and open in the Zariski topology. (Some authors prefer the term “general curve”, since the word “generic” has a more technical meaning in the theory of schemes.)

✖

§10. Real curves: The Harnack-Hilbert problem

In the real case, the most studied question about smooth curves in $\mathbb{P}^2(\mathbb{R})$ is the Hilbert-Harnack problem concerning the number and topological arrangement of the connected components, each of which is a topological circle. We provide a report about this problem, as well as a suggested reformulation. The paper concludes with a brief Appendix describing some of the literature on the moduli spaces that we consider.

2. Group actions and orbifolds

This section will provide a general introduction to quotient spaces under a smooth group action. In the best case, with a proper action, the quotient space is Hausdorff, with an orbifold structure. Since the group actions that we consider are not always proper, we also introduce a modified requirement of weakly proper action, which suffices to prove that the quotient is locally Hausdorff, with a weak orbifold structure which includes only some of the usual orbifold properties. The section will conclude by discussing the special case of the projective general linear group and its action on projective space.

First consider the complex case. Let $\mathbf{X}$ be a metrizable complex manifold, and let $\mathbf{G}$ be a complex Lie group⁠³ which acts on the left by a holomorphic map $\mathbf{G}\times \mathbf{X}\to \mathbf{X}$,

³

Although our Lie groups are always positive dimensional, the discussion would apply equally well to the case of a discrete group, which we might think of as a zero-dimensional Lie group.

✖

$$\begin{equation*} (\mathbf{g},\nobreakspace \mathbf{x})\nobreakspace \mapsto\nobreakspace \mathbf{g}(\mathbf{x}), \end{equation*}$$

where $\nobreakspace \mathbf{g}_1\big (\mathbf{g}_2(\mathbf{x})\big )=(\mathbf{g}_1\mathbf{g}_2)(\mathbf{x})$. (A manifold is metrizable if and only if it is paracompact and Hausdorff.) We will always assume that the action is effective in the sense that

$$\begin{equation*} \mathbf{g}(\mathbf{x})=\mathbf{x}\quad \text{for all }\nobreakspace\nobreakspace \mathbf{x}\quad \Longleftrightarrow \quad \mathbf{g}\quad \text{is the identity element}\nobreakspace\nobreakspace \mathbf{e}\in \mathbf{G}\,. \end{equation*}$$

The quotient space (or orbit space) in which $\mathbf{x}$ is identified with $\mathbf{x}\mathbf{'}$ if and only if $\mathbf{x}\mathbf{'}=\mathbf{g}(\mathbf{x})$ for some $\mathbf{g}$ will be denoted⁠⁴ by $\mathbf{X}/\mathbf{G}$.

⁴

Since $\mathbf{G}$ acts on the left, many authors would use the notation $\mathbf{G}{\setminus }\mathbf{X}$.

✖

In the real case, the definitions are completely analogous, although we could equally well work in either the $C^\infty$ category or in the real analytic category. To fix ideas, let us choose the real analytic category. Thus in the real case, we will assume that $\mathbf{G}$ is a real Lie group, that $\mathbf{X}$ is a metrizable real analytic manifold, and that $\mathbf{G}\times \mathbf{X}\to \mathbf{X}$ is a real analytic map. It will often be convenient to use the word analytic, by itself, to mean “real analytic” in the real case, or “complex analytic” in the complex case.

Definition 2.1.

For each $\mathbf{x}\in \mathbf{X}$ the set $\llparenthesis \mathbf{x}\rrparenthesis$ consisting of all images $\mathbf{g}(\mathbf{x})$ with $\mathbf{g}\in \mathbf{G}$ is called the $\mathbf{G}$-orbit of $\mathbf{x}$. We will also use the notation

$$\begin{equation*} \llparenthesis \mathbf{x}\rrparenthesis\nobreakspace =\nobreakspace \mathbf{F}\nobreakspace =\nobreakspace \mathbf{F}_\mathbf{x} \end{equation*}$$

if we are thinking of $\llparenthesis \mathbf{x}\rrparenthesis$ as a fiber of the projection map ${\boldsymbol{\pi }}:\mathbf{X}\to \mathbf{Y}=\mathbf{X}/\mathbf{G}$.

Remark 2.2.

In other words, each fiber is an equivalence class, where two points of $\mathbf{X}$ are equivalent if and only if they belong to the same orbit under the action of $\mathbf{G}$. More generally, given any equivalence relation ${\sim }$ on $\mathbf{X}$, we can form the quotient space $\mathbf{Y}=\mathbf{X}/\!{\sim }$. Such a quotient space $\mathbf{Y}$ always has a well-defined quotient topology, defined by the condition that a set $\mathbf{U}\subset \mathbf{Y}$ is open if and only if the preimage ${\boldsymbol{\pi }}^{-1}(\mathbf{U})$ is open as a subset of $\mathbf{X}$. (We will always assume that $\mathbf{X}$ is a Hausdorff space, but it does not necessarily follow that $\mathbf{Y}$ is Hausdorff.)

Remark 2.3 (Closed orbits and the $\mathrm{T}_1$ condition).

By definition, a topological space $\mathbf{Y}$ is a $\mathrm{T}_1$-space if every point of $\mathbf{Y}$ is closed as a subset of $\mathbf{Y}$. Evidently, a quotient space $\mathbf{X}/\mathbf{G}$ (or more generally $\mathbf{X}/\!{\sim }$) is a $\mathrm{T}_1$-space if and only if each orbit (or each equivalence class) is closed as a subset of $\mathbf{X}$.

Orbifolds and weak orbifolds.

The concept and basic properties of orbifolds are due to I. Satake, who called them V-manifolds (see Sa1Sa2). They were later studied by W. Thurston (see Th), who introduced the term “orbifold”.

Let $\mathbb{F}$ stand for either the real or the complex numbers.

Definition 2.4.

By a $d$-dimensional $\mathbb{F}$-orbifold chart around a point $\mathbf{y}$ of a topological space $\mathbf{Y}$ we will mean the following:

(1)

a finite group $\mathfrak{R}\subset \operatorname {GL}_d(\mathbb{F})$ acting linearly on $\mathbb{F}^d$;

(2)

an $\mathfrak{R}$-invariant open neighborhood $W$ of the point $\mathbf{0}\in \mathbb{F}^d$; and

(3)

a homeomorphism $h$ from the quotient space $W/\mathfrak{R}$ onto an open neighborhood $U$ of $\mathbf{y}$ in $\mathbf{Y}$ such that the zero vector in $W$ maps to $\mathbf{y}$.

The group $\mathfrak{R}$ will be called the ramification group at $\mathbf{y}$, and its order $\mathfrak{r}_\mathbf{y}\ge 1$ will be called the ramification index. A point $\mathbf{y}'$ is ramified if $\mathfrak{r}_{\mathbf{y}'}>1$ and unramified if $\mathfrak{r}_{\mathbf{y}'}=1$.

The space $\mathbf{Y}$ together with an integer valued function $\mathbf{y}\mapsto \mathfrak{r}_\mathbf{y}\ge 1$ will be called a $d$-dimensional weak orbifold over $\mathbb{F}$ if there exists such an orbifold chart $W/\mathfrak{R}\cong U$ around every point $\mathbf{y}\in \mathbf{Y}$, such that the associated ramification function from $U$ to the positive integers coincides with the specified function $\mathbf{y}'\mapsto \mathfrak{r}_{\mathbf{y}'}$ restricted to $U$.

Lemma 2.5.

Every weak orbifold is a locally Hausdorff space; and the function $\mathbf{y}\mapsto \mathfrak{r}_\mathbf{y}\ge 1$ from $\mathbf{Y}$ to the set of positive integers is always upper semicontinuous, taking the value $\mathfrak{r}_\mathbf{y}=1$ on a dense open set. More precisely, for any orbifold chart $W/\mathfrak{R}\stackrel{\cong }{\longrightarrow } U=U_\mathbf{y}$, we have $\mathfrak{r}_{\mathbf{y}'}\le \mathfrak{r}_\mathbf{y}$ for every $\mathbf{y}'\in U$, with $\mathfrak{r}_{\mathbf{y}'}=1$ on a dense open subset of $U$.

Of course in good cases $\mathbf{Y}$ will be a Hausdorff space; but even a locally Hausdorff space can be quite useful.⁠⁵

⁵

Perhaps the most startling application of locally Hausdorff spaces in science would be to the “Many Worlds” interpretation of quantum mechanics, in which the space-time universe continually splits into two or more alternate universes. (See, for example, Bec.) The resulting object is possibly best described as a space which is locally Hausdorff, but wildly non-Hausdorff. It can be constructed mathematically out of infinitely many copies of the Minkowski space $\mathbb{R}^{3,1}$ by gluing together corresponding open subsets. (Of course it does not make any objective sense to ask whether these alternate universes “really exist”. The only legitimate question is whether a mathematical model including such alternate universes can provide a convenient and testable model for the observable universe.)

✖

Proof of Lemma 2.5.

For any chart $W/\mathfrak{R}\stackrel{\cong }{\longrightarrow } U$ around $\mathbf{y}$ and any nonidentity element $\mathbf{g}\in \mathfrak{R}$, the set of elements of $\mathbb{F}^d$ fixed by $\mathbf{g}$ must be a linear subspace of $\mathbb{F}^d$ of dimension at most $d-1$. The complement of this finite union of linear subspaces within $W$ is a dense open subset $W_0\subset W$. If $U_0\cong W_0/\mathfrak{R}$ is its image, then the associated mapping $W_0\to U_0$ is locally bijective, and it is precisely $\mathfrak{r}_\mathbf{y}$-to-one. Since this map is one-to-one in a small neighborhood of any point of $W_0$, it follows that all points of $U_0$ are unramified. Each $\mathfrak{R}$-orbit in $U$ is compact and nonempty, so we can use the Hausdorff metric for compact subsets of $\mathbb{F}^d$ to make $U\cong W/\mathfrak{R}$ into a metric space. In particular, it follows that $\mathbf{Y}$ is locally Hausdorff.

■

Definition 2.6.

An orbifold chart $W/\mathfrak{R}\cong U$ around $\mathbf{y}$ gives rise to a smaller orbifold chart around any point $\mathbf{y}'\in U$ which is called the restriction of this chart to a neighborhood of $\mathbf{y}'$. In fact, choosing a representative point $\mathbf{w}'\in W$ over $\mathbf{y}'$, let $\mathfrak{R}_{\mathbf{w}'}$ be the stabilizer, consisting of all $\mathbf{g}\in \mathfrak{R}$ for which $\mathbf{g}(\mathbf{w}')=\mathbf{w}'$. Evidently $\mathfrak{R}_{\mathbf{w}'}$ acts linearly on $W$, fixing the point $\mathbf{w}'$. Choose an $\mathfrak{R}_{\mathbf{w}'}$-invariant neighborhood $W'$ of $\mathbf{w}'$ which is small enough so that the various images $\mathbf{g}(W')$ with $\mathbf{g}\in \mathfrak{R}{\smallsetminus }\mathfrak{R}_{\mathbf{w}'}$ are all disjoint from $W'$. Then the projection from $W'$ to $W'/\mathfrak{R}_{\mathbf{w}'}\subset U$ is the required restriction to an orbifold chart around $\mathbf{y}'$.

Definition 2.7.

An orbifold atlas on $Y$ is a collection of orbifold chart homeomorphisms

$$\begin{equation*} h_j: W_j/\mathfrak{R}_j\stackrel{\cong }{\longrightarrow } U_j\subset Y, \end{equation*}$$

where the $U_j$ are open sets covering $Y$, which satisfy the following.

Compatibility condition. For each point $\mathbf{z}$ in an overlap $U_i\cap U_j$ and each sufficiently small neighborhood $U'$ of $\mathbf{z}$, the restriction of the $i$th and $j$th orbifold charts to $U'$ are isomorphic in the following sense. Let

$$\begin{equation*} h'_i: W'_i/\mathfrak{R}_{i,w_i}\stackrel{\cong }{\longrightarrow } U'\qquad \text{and} \qquad h'_j:W'_j/\mathfrak{R}_{j,w_j}\stackrel{\cong }{\longrightarrow } U' \end{equation*}$$

be the two restrictions. Then we require that there should be an analytic isomorphism $\nobreakspace \psi :W'_i\stackrel{\cong }{\longrightarrow } W'_j\nobreakspace$ so that $h'_i=h'_j\circ \psi$. (Compare Figure 1.) Furthermore, there should be a group isomorphism $\nobreakspace \phi :\mathfrak{R}_{i, w_i}\stackrel{\cong }{\longrightarrow }\mathfrak{R}_{j,w_j}\nobreakspace$ so that

$$\begin{equation*} \psi \big (\mathbf{g}(w)\big )\nobreakspace =\nobreakspace \phi (\mathbf{g})\big (\psi (w)\big ) \end{equation*}$$

for every $\mathbf{g}\in \mathfrak{R}_{i, w_i}$ and every $w\in W'_i$.

Two such atlases are equivalent if their union also satisfies this compatibility condition. The space $\mathbf{Y}$ together with an equivalence class of such atlases is called an orbifold.⁠⁶ (Compare Th, BMP.)

⁶

Caution. Most authors require orbifolds to be Hausdorff spaces, but as noted in the Introduction, we will allow orbifolds which are only locally Hausdorff.

✖

There may well be examples of weak orbifolds which are not compatible with any orbifold structure, but we do not know any such examples.

Example 2.8.

On any Riemann surface, we can choose any function $\mathbf{y}\mapsto \mathfrak{r}_\mathbf{y}\ge 1$ which takes the value $\mathfrak{r}_\mathbf{y}=1$ except at finitely many points. A corresponding collection of complex orbifold charts is easily constructed, and the compatibility condition is easily verified.

Here is a pair of more interesting examples.

Example 2.9.

Let $\mathbf{X}\subset \mathbb{R}^3$ be the plane consisting of all $(x,\,y,\,z)$ with $x+y+z=0$, and let $\mathfrak{G}_3$ be the group of permutations of the three coordinates. Then under the action of $\mathfrak{G}_3$, each point of $\mathbf{X}$ can be put into a unique normal form with $x\le y\le z$. If we set $u=y-x\ge 0$ and $v=z-y\ge 0$, then we can solve easily for $x,\,y,\,z$ as linear functions of $u,\,v$. Thus the quotient space $\mathbf{Y}=\mathbf{X}/\mathfrak{G}_3$ can be identified with the positive quadrant in the $(u,\,v)$-plane. The ramification index is $\mathfrak{r}(u,\,v)=1$ throughout the interior of the quadrant, with $\mathfrak{r}(u,\,0)=\mathfrak{r}(0,\,v)=2$ along the two edges, and with $\mathfrak{r}(0,\,0)=6$ at the origin. Note that the compatibility is automatically satisfied, since we have specified only one coordinate chart.

The analogous example with $(x,\,y,\,z)\in \mathbb{C}^3$ looks quite different. In this case, the quotient space is best described as the complex $(\sigma _2,\,\sigma _3)$-plane, where

$$\begin{equation*} \sigma _2=xy+yz+zx\qquad \text{and}\qquad \sigma _3=xyz \end{equation*}$$

are elementary symmetric functions, with $\sigma _1=x+y+z=0$. Again the ramification index of a point in the quotient space is $\mathfrak{r}=1$ if $x,\,y,\,z$ are all distinct, $\mathfrak{r}=2$ if only two are equal to each other, and $\mathfrak{r}=6$ if $x=y=z=0$. Any triple with $\mathfrak{r}=2$ takes the form $(x,\,x,\,-2x)$, up to permutation of the coordinates, with $\sigma _2=-3x^2$ and $\sigma _3=-2x^3$. Thus $\mathfrak{r}=1$ if the expression⁠⁷ $\nobreakspace 4\sigma _2^3+27\sigma _3^2\nobreakspace$ is nonzero; with $\mathfrak{r}=2$ for most points where this expression is zero; but with $\mathfrak{r}=6$ if $\sigma _2=\sigma _3=0$.

⁷

Up to sign, this expression is just the discriminant of the polynomial $t^3+\sigma _2 t-\sigma _3= (t-x)(t-y)(t-z)$.

✖

The main object of this section will be to describe conditions on the group action which guarantee that the quotient will be an orbifold or weak orbifold.

Proper and weakly proper actions.

Definition 2.10.

A continuous action of $\mathbf{G}$ on a locally compact space $\mathbf{X}$ is called proper if the associated map

$$\begin{equation*} (\mathbf{g},\,\mathbf{x})\nobreakspace \mapsto\nobreakspace \big (\mathbf{g}(\mathbf{x}),\,\mathbf{x}\big ) \end{equation*}$$

from $(\mathbf{G}\times \mathbf{X})$ to $(\mathbf{X}\times \mathbf{X})$ is a proper map (in the usual sense that the preimage of any compact set is compact). A completely equivalent requirement is the following.

For any pair of compact subsets $K,\,K'\subset \mathbf{X}$ the set of all $\mathbf{g}$ with $\mathbf{g}(K)\cap K'\ne \emptyset$ is compact.

Here is another completely equivalent condition.

For every pair of points $\mathbf{x}$ and $\mathbf{x}\mathbf{'}$ in $\mathbf{X}$, there exist neighborhoods $U$ of $\mathbf{x}$ and $U'$ of $\mathbf{x}\mathbf{'}$ which are small enough so that the set of all $\mathbf{g}\in \mathbf{G}$ with $\mathbf{g}(U)\cap U'\ne \emptyset$ has compact closure.

The proof that these three forms of the definition are equivalent is straightforward and will be left to the reader.⁠⁸

⁸

In the special case of a discrete group, such an action is called properly discontinuous. (For more about proper actions, see the discussion beginning with Lemma 2.25.)

✖

The action is locally proper at $\mathbf{x}$ if this condition is satisfied for the special case where $\mathbf{x}=\mathbf{x}\mathbf{'}$. It then follows that the action is proper throughout some $\mathbf{G}$-invariant open neighborhood of $\mathbf{x}$. In fact, if each $\mathbf{g}$ which maps some point of $U$ into $U$ is contained in a compact set $K\subset \mathbf{G}$, then it follows that each $\mathbf{g}$ which maps a point of $\mathbf{g}_1(U)$ to a point of $\mathbf{g}_2(U)$ is contained in the compact set $\mathbf{g}_2 K\mathbf{g}_1^{-1}\subset \mathbf{G}$.

The action will be called weakly proper at $\mathbf{x}$ if the following still weaker local condition is satisfied:

There should be a neighborhood $U$ of $\mathbf{x}$ and a compact set $K\subset \mathbf{G}$ such that, whenever two points $\mathbf{x}_1$ and $\mathbf{x}_2$ of $U$ belong to the same $\mathbf{G}$-orbit, there exists at least one element $\mathbf{g}\in K$ with $\mathbf{g}(\mathbf{x}_1)=\mathbf{x}_2$.

It will be convenient to call a point in $\mathbf{X}/\mathbf{G}$ either proper or improper according as the action of $\mathbf{G}$ on corresponding points of $\mathbf{X}$ is or is not locally proper. Similarly, an improper point in $\mathbf{X}/\mathbf{G}$ will be called weakly proper if the action of $\mathbf{G}$ on corresponding points of $\mathbf{X}$ is weakly proper. (Caution. Even when a point is improper, the quotient space may have a perfectly good orbifold structure.)

Definition 2.11.

Given an action of $\mathbf{G}$ on $\mathbf{X}$, the stabilizer $\mathbf{G}_\mathbf{x}$ of a point $\mathbf{x}\in \mathbf{X}$ is the closed subgroup of $\mathbf{G}$ consisting of all $\mathbf{g}\in \mathbf{G}$ for which $\mathbf{g}(\mathbf{x})=\mathbf{x}$. Note that points on the same fiber have isomorphic stabilizers, since

$$\begin{equation*} \mathbf{G}_{\mathbf{g}(\mathbf{x})}\nobreakspace =\nobreakspace \mathbf{g}\, \mathbf{G}_\mathbf{x}\mathbf{g}^{-1}. \end{equation*}$$

If the stabilizer $\mathbf{G}_\mathbf{x}$ is finite, then it follows easily that the fiber $\mathbf{F}$ through $\mathbf{x}$ (consisting of all images $\mathbf{g}(\mathbf{x})$ with $\mathbf{g}\in \mathbf{G}$) is a smoothly embedded submanifold which is locally diffeomorphic to $\mathbf{G}$.

Under the hypothesis that all stabilizers are finite, we will prove the following in Theorem 2.18 together with Lemma 2.13 and Corollary 2.28:

•

For a proper action the quotient space is a Hausdorff orbifold.

•

For a locally proper action the quotient is a locally Hausdorff orbifold.

•

For a weakly proper action the quotient is a locally Hausdorff weak orbifold.

(See Figure 2 for an example of a smooth locally proper action with trivial stabilizers where the quotient is not a Hausdorff space.)

Remark 2.12.

Although there are examples which are weakly proper but not locally proper, they seem to be hard to find. Lemma 3.11 will show that divisors of degree four with only three distinct points give rise to such examples; and the proof of Lemma 5.4 will show that curves of degree three with a simple double point provide closely related examples. However, these are the only examples we know.

The following is well known.

Lemma 2.13.

If the action is proper, then the quotient $\mathbf{X}/\mathbf{G}$ is a Hausdorff space.

It follows as an immediate corollary that a locally proper action yields a quotient space which is locally Hausdorff. However the quotient under a locally proper action need not be Hausdorff. (Compare Figure 2.) If stabilizers are finite, then we will see in Theorem 2.18 that even a weakly proper action yields a quotient space which is locally Hausdorff.

Remark 2.14.

Note that every locally Hausdorff space is $\mathrm{T}_1$. In fact if one point $\mathbf{p}$ belongs to the closure of a different point $\mathbf{q}$, then no neighborhood of $\mathbf{p}$ is Hausdorff.

Proof of Lemma 2.13.

It will be convenient to choose a metric on $\mathbf{X}$. Given $\mathbf{x}$ and $\mathbf{x}\mathbf{'}$ there are two possibilities. If we can choose neighborhoods $U$ and $U'$ so that no translate $\mathbf{g}(U)$ intersects $U'$, then the images ${\boldsymbol{\pi }}(U)$ and ${\boldsymbol{\pi }}(U')$ in the quotient space are disjoint open sets.

On the other hand, taking $U_j$ and $U'_j$ to be a sequence of neighborhoods of $\mathbf{x}$ and $\mathbf{x}\mathbf{'}$, respectively, of radius $1/j$, if we can choose a group element $\mathbf{g}_j$ for each $j$ with $\mathbf{g}_j(U_j)\cap U'_j\ne \emptyset$, then by compactness we can pass to an infinite subsequence so that the $\mathbf{g}_j$ converge to a limit $\mathbf{g}$. It follows easily that $\mathbf{g}(\mathbf{x})=\mathbf{x}\mathbf{'}$, so that $\mathbf{x}$ and $\mathbf{x}\mathbf{'}$ map to the same point in the quotient space.

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Remark 2.15.

The converse to Lemma 2.13 is false: A quotient space may be Hausdorff even when the action is not proper. Compare the discussions of $\mathfrak{M}_4(\mathbb{C})$ and $\mathfrak{M}_4(\mathbb{R})$ in Lemmas 3.4, 3.5, and 3.11, and of the quotient spaces $\mathbb{M}_3(\mathbb{C})$ and $\mathbb{M}_3(\mathbb{R})$ in Section 5. These quotients are Hausdorff orbifolds, even though the associated group action fails to be proper everywhere. (See Example 2.8, as well as Lemma 5.4.) However, for larger $n$ we will have to deal with moduli spaces which are definitely not Hausdorff.  (See Theorem 3.2 for $\mathfrak{M}_n$ and Theorem 6.1 for $\mathbb{M}_n$.)

Weak orbifold structures.

The passage from weakly proper actions to weak orbifold structures will be based on the following. Given any fiber $\mathbf{F}$, and given any point $\mathbf{x}\in \mathbf{F}$, we will refer to the quotient of tangent vector spaces

$$\begin{equation*} V_\mathbf{x}\nobreakspace =\nobreakspace T_\mathbf{x}\mathbf{X}/T_\mathbf{x}\mathbf{F} \end{equation*}$$

as the transverse vector space to $\mathbf{F}$ at $\mathbf{x}$. (If $\mathbf{X}$ is provided with a Riemannian metric, then $V_\mathbf{x}$ can be identified with the normal vector space at $\mathbf{x}$.)

In order to describe a weak orbifold structure on the quotient, we must first construct the associated ramification groups. Note that the stabilizer $\mathbf{G}_\mathbf{x}$ acts linearly on both $T_\mathbf{x}\mathbf{X}$ and $T_\mathbf{x}\mathbf{F}$, and hence acts linearly on the $d$-dimensional quotient space $V_\mathbf{x}$, where $d$ is the codimension of $\mathbf{F}$ in $\mathbf{X}$. In practice, we will always assume that the stabilizer is finite, so that $d$ is equal to the difference $\dim (\mathbf{X})-\dim (\mathbf{G})$. However, this action is not always effective: the group $\mathbf{G}_\mathbf{x}$ may act nontrivially on $T_\mathbf{x}\mathbf{F}$, while leaving the transverse vector space pointwise fixed.

Definition 2.16.

Let $\mathbf{H}_\mathbf{x}$ be the normal subgroup of $\mathbf{G}_\mathbf{x}$ consisting of all group elements which act as the identity map on $V_\mathbf{x}$ (that is, all $\mathbf{h}\in \mathbf{G}_\mathbf{x}$ such that $\mathbf{h}(\mathbf{v})=\mathbf{v}$ for all $\mathbf{v}\in V_\mathbf{x}$). The quotient group

$$\begin{equation*} \nobreakspace {\mathfrak{R}}_\mathbf{x}\nobreakspace =\nobreakspace \mathbf{G}_\mathbf{x}/\mathbf{H}_\mathbf{x}\nobreakspace \end{equation*}$$

will be called the ramification group at $\mathbf{x}$. Note that by its very definition, ${\mathfrak{R}}_\mathbf{x}$ comes with a linear action on the vector space $V_\mathbf{x}$, which is isomorphic to $\mathbb{R}^d$ or $\mathbb{C}^d$. It is not hard to check that different points on the same fiber $\mathbf{F}$ have isomorphic ramification groups. Let $\mathbf{y}$ be the image of the fiber $\mathbf{F}$ in $\mathbf{Y}$. As in Definition 2.4, the order of this finite group $\mathfrak{R}_\mathbf{x}$ will be called the ramification index $\mathfrak{r}=\mathfrak{r}(\mathbf{y})\ge 1$, and $\mathbf{y}$ will be called unramified if $\mathfrak{r}(\mathbf{y})=1$.

We will need the following.

Lemma 2.17 (Invariant metrics).

In the real case, given any finite subgroup $\Gamma \subset \mathbf{G}$ there exists a smooth $\Gamma \!$-invariant Riemannian metric on the space $\mathbf{X}\,.$ Similarly, in the complex case $\mathbf{X}$ has a smooth $\Gamma \!$-invariant Hermitian metric.

Proof.

Starting with an arbitrary smooth Riemannian or Hermitian metric, average over its transforms⁠⁹ by elements of $\Gamma$. Then each element of $\Gamma$ will represent an isometry for the averaged metric.

⁹

A Riemannian metric can be described as a smooth function $\mu$ which assigns to each $\mathbf{x}\in \mathbf{X}$ a symmetric positive definite inner product $\mu _\mathbf{x}(v,w)$ on the vector space $T_\mathbf{x}\mathbf{X}$ of tangent vectors at $\mathbf{x}$. Given any diffeomorphism $f:\mathbf{X}\to \mathbf{X}\mathbf{'}$, and given a Riemannian metric $\mu$ on $\mathbf{X}\mathbf{'}$, we can use the first derivative map $f_*:T_\mathbf{x}\mathbf{X}\stackrel{\cong }{\longrightarrow }T_{f(\mathbf{x})}\mathbf{X}\mathbf{'}$ to pull back the metric, setting

$$\begin{equation*} (f^*\mu )_\mathbf{x}(v,w)\nobreakspace =\nobreakspace \mu _{f(\mathbf{x})}\big (f_*(v),\,f_*(w)\big )\nobreakspace \in\nobreakspace \mathbb{R}. \end{equation*}$$

In particular, given any finite group $\Gamma$ consisting of $r$ diffeomorphisms from $\mathbf{X}$ to itself, we can form the average $\widehat{\mu }\nobreakspace =\nobreakspace \frac{1}{r}\sum _{\mathbf{g}\in \Gamma } \mathbf{g}^*\mu .$ The construction in the complex case is similar, using Hermitian inner products.

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Now let $\mathbf{F}$ be any fiber with finite stabilizers, and let $\mathbf{x}_0\in \mathbf{F}$ be an arbitrary base point. Since the stabilizer $\mathbf{G}_{\mathbf{x}_0}$ is a finite group acting on $\mathbf{X}$, we can choose a $\mathbf{G}_{\mathbf{x}_0}\!$-invariant metric on $X$. Using this metric, the transverse vector space $V_{\mathbf{x}_0}\nobreakspace =\nobreakspace T_{\mathbf{x}_0} \mathbf{X}/T_{\mathbf{x}_0}\mathbf{F}$ can be identified with the normal vector space consisting of all tangent vectors to $\mathbf{X}$ at $\mathbf{x}_0$ which are orthogonal to the fiber at ${\mathbf{x}_0}$. Given $\varepsilon >0$, we can consider geodesics of length $\varepsilon$ starting at $\mathbf{x}_0$ which are orthogonal to $\mathbf{F}$ at $\mathbf{x}_0$. If $\varepsilon$ is small enough, these geodesics will sweep out a smooth $d$-dimensional disk $D_\varepsilon$ which meets $\mathbf{F}$