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.