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Group Actions, Divisors, and Plane Curves BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 57, Number 2, April 2020, Pages 171–267 https://doi.org/10.1090/bull/1681 Article electronically published on February 7, 2020 GROUP ACTIONS, DIVISORS, AND PLANE CURVES ARACELI BONIFANT AND JOHN MILNOR 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 P1, modulo the group of projective transformations of P1; and then the moduli space of curves (or more generally effective algebraic 1-cycles) with finite stabilizer in P2, modulo the group of projective transformations of P2. It also discusses automorphisms of curves and the topological classification of smooth real curves in P2. Contents 1. Introduction 171 2. Group actions and orbifolds 174 1 3. Divisors on P and the moduli space Mn 191 4. Curves (or 1-cycles) in P2 and their moduli space 208 5. Cubic curves 212 6. Degree at least four 221 7. Singularity genus and proper action 230 8. Infinite automorphism groups 239 9. Finite automorphism groups 247 10. Real curves: The Harnack-Hilbert problem 257 Appendix A. Remarks on the literature 262 Acknowledgments 263 About the authors 263 References 263 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 Received by the editors March 11, 2019. 2010 Mathematics Subject Classification. Primary 14L30, 14H50, 57R18, 14H10; Secondary 08A35, 14P25. Key words and phrases. Moduli space of curves, smooth complex curves, effective 1-cycles, stabilizers of curves, algebraic set, group actions, proper action, improper action, weakly proper action, orbifolds, weak orbifolds, rational homology manifold, tree-of-spheres, W-curves, moduli space of divisors, Deligne-Mumford compactification. The first author wants to thank the Institute for Mathematical Sciences at Stony Brook Uni- versity, where she spent her sabbatical year, for its support to this project. c 2020 American Mathematical Society 171 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 172 ARACELI BONIFANT AND JOHN MILNOR 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 intro- duce the concepts of locally proper action,andweakly 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: 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. 1 §3. Divisors on P and the moduli space Mn. Working either over the real or complex numbers, the space Mn of effective divisors of degree n with finite stabilizer 1 on the projective line P , modulo projective transformations, is a T1 space for every n, but it is a Hausdorff orbifold only for n ≤ 4. In the complex case the space M4(C) 1 is homeomorphic to P (C), but in the real case M4(R) is homeomorphic to a closed line segment in P1(R). In both the real and complex cases, there is just one point of M4 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 Haus ⊂ n>4, there is a unique maximal open subset Mn Mn which is a Hausdorff Haus orbifold, but Mn is not even locally Hausdorff at points outside of Mn .This Haus subset Mn is compact for n odd, but not for n even. The proofs make use of a Distortion Lemma for automorphisms of P1 which describes the way in which an automorphism that lies outside a large compact sub- set of PGL2 must distort the geometry. The proofs also make use of two familiar projective invariants associated with a 4-tuple of points in P1. The cross-ratio de- pends on the ordering of the four points, while the shape invariant J = J(x, y, z, w) (the classical j-invariant times a convenient constant) is invariant under permuta- tions of the four points. In the complex case, a close relative of our Mn(C) is the classical moduli space M0,n consisting of closed Riemann surfaces of genus zero provided with an ordered list of n ≥ 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 M0,n(C), a beautiful object introduced by Knudsen, based on ideas of Grothendieck, Deligne, and Mumford. (See also Keel [Ke] and Etingof, M R Henriques, Kamnitzer, and Rains [EHKR] for the analogous space 0,n( ).) In n both the real and complex cases, there are associated embeddings p M0,p+1 × M0,q+1 → M0,n, 1Most authors require orbifolds to be Hausdorff spaces, however we will allow orbifolds which are only locally Hausdorff . License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use GROUP ACTIONS, DIVISORS, AND PLANE CURVES 173 where p + q = n. Asanexample,intherealcaseforp =2andq =3,there are ten essentially different embeddings of the circle M0, 3 × M0, 4 into the surface M0, 5, cutting it up into 12 hyperbolic pentagons, and thus presenting it as a hy- perbolic analogue of the dodecahedron. The space M0, 6(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 corre- sponding to the embeddings of M0, 4 × M0, 4 into M0, 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 P2 and their moduli space. The space of all curves of degree n in P2 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 P2. These are called effective (algebraic) 1-cycles. For n ≥ 3, we study the moduli space Mn consisting of all projective equivalence classes of effective 1-cycles 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 P2 but also divisors in P1.) However, our presentation is more elementary, and is addressed to nonspecialists. §5. Cubic curves. The space M3(C) is homeomorphic to the topological 2-sphere consisting of all ratios (a3 : b2) ∈ P1(C), corresponding to curves y2 = x3 + ax + b. It is isomorphic as an orbifold to the moduli space M4(C) for divisors of degree 1 four on P (C). In the real case, the space M3(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 y2 = x3 + ax + b with a2 + b2 = 1, but also in terms of the Hesse normal form x3 + y3 + z3 =3kxyz, with k ∈ R ∪{∞},andin terms of a convenient flex-slope normal form, parametrized by the slope s at a flex point. §6. Degree at least four. When n ≥ 5, we prove that the moduli space Mn(C)is not a Hausdorff space. On the other hand, we provide two different procedures for describing large open subsets of Mn(C) which are Hausdorff orbifolds. One of them is based on studying the distribution of virtual flex points,thatispointswhichare either flex points or which yield flex points under perturbation. The other will be discussed in §7. Msm ⊂ M C P2 C The open subset n n( ) corresponding to smooth curves in ( ) is a Hausdorff space, which maps naturally into the classical moduli space Mg(n) consisting of all isomorphism classes of compact Riemann surfaces of genus g(n)= n−1 ≥ 2 . This map is injective for all n 3 (compare Chang [Ch]), but is an isomor- phism only for n =3. §7. Singularity genus and proper action. This section provides a different proof that large subsets of Mn(C) are Hausdorff orbifolds, based on the genus invariant for singular points. Both proofs make use of a Distortion Lemma for automorphisms of P2 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. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 174 ARACELI BONIFANT AND JOHN MILNOR §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.
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