# 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 modulo the group of projective transformations of , and then the moduli space of curves (or more generally effective algebraic ; with finite stabilizer in -cycles) modulo the group of projective transformations of , It also discusses automorphisms of curves and the topological classification of smooth real curves in . .

## 1. Introduction

A basic objective of this paper is to provide an elementary introduction to the moduli space of curves of degree 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

**. We introduce the usual concept of orbifold,Footnote**

*weakly locally proper action*^{1}as well the concept of

**for quotient spaces which have only some of the standard orbifold properties, and we prove the following:**

*weak orbifold*^{1}

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

**.**

*locally Hausdorff*For a

proper/locally proper/orweakly properaction the quotient space is aHausdorff orbifold/locally Hausdorff orbifold/orlocally Hausdorff weak orbifold, respectively.

### §3. Divisors on and the moduli space .

Working either over the real or complex numbers, the space of effective divisors of degree with finite stabilizer on the projective line modulo projective transformations, is a , space for every but it is a Hausdorff orbifold only for , In the complex case the space . is homeomorphic to but in the real case , is homeomorphic to a closed line segment in In both the real and complex cases, there is just one point of . 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 there is a unique maximal open subset , which is a Hausdorff orbifold, but is not even locally Hausdorff at points outside of This subset . is compact for odd, but not for even.

The proofs make use of a ** Distortion Lemma** for automorphisms of which describes the way in which an automorphism that lies outside a large compact subset of must distort the geometry. The proofs also make use of two familiar projective invariants associated with a 4-tuple of points in The cross-ratio depends on the ordering of the four points, while the shape invariant . (the classical times a convenient constant) is invariant under permutations of the four points. -invariant

In the complex case, a close relative of our is the classical moduli space consisting of closed Riemann surfaces of genus zero provided with an ordered list of 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 a beautiful object introduced by Knudsen, based on ideas of Grothendieck, Deligne, and Mumford. (See also Keel ,Reference Ke and Etingof, Henriques, Kamnitzer, and Rains Reference EHKR for the analogous space

where

### §4. Curves (or -cycles) in and their moduli space

The space of all curves of degree ** effective (algebraic) **. For

### §5. Cubic curves

The space *flex-slope normal form*, parametrized by the slope

### §6. Degree at least four.

When *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

### §7. Singularity genus and proper action

This section provides a different proof that large subsets of

### §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

### §9. Finite automorphism groups.

The study of finite automorphism groups of curves is closely related to the study of the moduli space ^{2} 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

## 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 ^{3} which acts on the left by a holomorphic map

^{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.

where ** effective** in the sense that

The ** quotient space** (or orbit space) in which

^{4}by

In the real case, the definitions are completely analogous, although we could equally well work in either the

### Orbifolds and weak orbifolds.

The concept and basic properties of orbifolds are due to I. Satake, who called them V-manifolds (see Reference Sa1Reference Sa2). They were later studied by W. Thurston (see Reference Th), who introduced the term “orbifold”.

Let

Of course in good cases ^{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, Reference 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

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

Here is a pair of more interesting examples.

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.

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.)

The following is well known.

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.

### Weak orbifold structures.

The passage from weakly proper actions to weak orbifold structures will be based on the following. Given any fiber

as the ** transverse vector space** to

In order to describe a weak orbifold structure on the quotient, we must first construct the associated ramification groups. Note that the stabilizer

We will need the following.

Now let