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One-Dimensional Substacks of the Moduli Stack of Deligne-Mumford Stable Curves

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One-Dimensional Substacks of the Moduli Stack of Deligne-Mumford Stable Curves One-dimensional substacks of the moduli stack of Deligne-Mumford stable curves Dr. rer. nat. J¨org Zintl arXiv:math/0612802v1 [math.AG] 27 Dec 2006 Habilitationsschrift (rev.) Kaiserslautern 2005 3 Abstract It is well known that there is a stratification of the moduli space of Deligne- Mumford stable curves of genus g by subschemes (0) (3g−3) M g = Mg ∪ . ∪ Mg , (i) where each Mg is of codimension i, and defined as the locus of stable (3g−4) curves with precisely i nodes. A connected component D of Mg can be identified with a scheme M ′, which is the moduli space of a moduli stack M′ of certain pointed stable curves of some genus g′ ≤ g. Our objects of study are those one-dimensional substacks D of the moduli stack Mg of Deligne-Mumford stable curves of genus g, which are defined as reductions of the preimages of components D in the above stratification. Even though there is an isomorphism M ′ =∼ D, we find that the stacks D and M′ are in general not isomorphic. We construct explicitely a surjective morphism on dense and open substacks M× −→ D×, and show that is representable, unramified and finite of degree equal to the order of a certain group of automorphisms. Using this, we can give a new description of the substacks D generically as global quotients. We explain how the above morphism can be extended to the stacks corresponding to the natural compactifications of M ′ and D, and we discuss when and how the quotient description of D extends. Finally, it is indicated how the results can be generalized to substacks defined by boundary strata of codimension less than 3g−4. The main theorem, which relates the stack D to a quotient of a substack of a moduli stack of pointed stable curves of genus less or equal to g, carries over immediately. In an appendix, an extensive compendium on stacks is provided, with a par- ticular focus on quotient stacks, including a few as yet unpublished observa- tions. 4 Contents 0 Introduction 7 1 Motivation 11 2 Moduli of pointed stable curves 17 3 The moduli problem 35 4 The moduli substack 73 5 Partial compactification 93 6 Glueing 111 7 The global picture 129 8 Examples and Outlook 145 8.1 Thecaseofgenusthree . 145 5 6 CONTENTS 8.2 Higher-dimensional boundary strata . 151 8.3 A remark on intersection products . 161 A Proof of lemma 3.42 165 B Stacks and Quotient Stacks 183 B.1 AbstractPreliminaries . 184 B.2 Stacks ...............................196 B.2.1 Descent ..........................196 B.2.2 Representability . 200 B.2.3 Algebraic Stacks . 204 B.2.4 Properties of Deligne-Mumford stacks . 208 B.3 Quotients .............................224 B.3.1 Groupactions. 224 B.3.2 Quotients of schemes . 227 B.3.3 Principalbundles. 229 B.3.4 Quotient fibred categories . 235 B.3.5 Quotient stacks . 259 List of Notations 279 Bibliography 283 Chapter 0 Introduction The study of moduli spaces often turns out to be technically demanding. At the same time, these challenges may lead to the developement of intricate and intriguing new theories. The theory of geometric invariants is just one example of many. Algebraic stacks are another instance of highly sophisticated techniques to approach moduli problems. They prove to be increasingly effective, as many questions on moduli can naturally be formulated in the language of cate- gories and functors. Often answers turn out to be much simpler, when phrased in terms of stacks, than they were in terms of schemes. At least, this is true formally. The simplification gained by the use of these tools is paid for with an increased effort necessary to built up the formal framework. As J. Harris and I. Morrison put it: “the initial learning curve is steep...”, [HM, p. 140]. What one gets in the end is a powerful instrument indeed, but also a concept which is less intuitively manageable than the concept of schemes, say. The starting point of our work is analyse one instance, where stacks do not behave in the way one would expect them to from looking at the corresponding moduli schemes only. This is a technical paper, but I am starting with a straightforward geomet- rical question. Consider the moduli space M g of Deligne-Mumford stable curves of a given genus g. Contained in it is a subscheme D as the locus of stable curves sharing a certain property. Suppose that this property can 7 8 CHAPTER 0. INTRODUCTION be formulated as a second moduli problem of its own right, with a moduli space M ′ isomorphic to D. Now compare the two stacks, one defined as the reduced substack D of the moduli stack Mg of Deligne-Mumford stable curves, which lies over D, the other as the stack M′ corresponding to the second moduli problem. What is the relation of D and M′? In general, the two stacks will not be isomorphic. The moduli stack D still “parametrizes” those features of objects represented by points of D, which do not vary within D. In our case, it will turn out that D and M′ differ generically by the action of a finite group A, which acts trivially on the moduli space M ′. While there is obviously an isomorphism between the schemes M ′ and M ′/A, the stacks M′, M′/A and D may be different. Here, the intuition aquired from working with schemes can no longer be used as a guide. This work is organized as follows. The first chapter illustrates in an example a question asked by A. Hirschowitz. Fix a stable curve C0 of some given genus g, which has the maximal number of nodes that is possible, and fix one of its nodes P1. Now look at such deformations of C0, which preserve all nodes except P1. This distinguishes an one-dimensional substack D(C0; P1) of the moduli stack Mg of all stable curves of genus g. What can be said about D(C0; P1)? In the second chapter, I recall some basic definitions and facts about pointed stable curves, as far as I need them later on. This includes a slight variation of Edidin’s theorem [Ed] about the quotient representation of the moduli stack of Deligne-Mumford stable curves. I also present the notion of an m/Γ-pointed stable curve of genus g, where Γ is a subgroup of the per- mutation group on m elements. Such a curve should be thought of as an m-pointed stable curve, but where the labels of its marked points are deter- mined only up to permutations in Γ. The actual definiton 2.14 is somewhat more technical to get the behaviour of curves in families right. (3g−4) In chapter 3, the basic objects of study are introduced. The locus M g of stable curves of genus g with at least 3g − 4 nodes is a one-dimensional subscheme of M g, which decomposes into several irreducible components. To specify one such component, I am fixing one stable curve C0 with 3g − 3 nodes, together with one node P1 on it. Deformations of C0, preserving all nodes but P1, distinguish a unique irreducible component D(C0; P1) of 9 (3g−4) M g . After a characterization of those stable curves, which are rep- resented by points of D(C0; P1), a description of D(C0; P1) in terms of a ′ moduli space M g+,m of certain pointed stable curves is given. This charac- terization leads also to the definition of a substack D(C0; P1) of the stack Mg, which has D(C0; P1) as its moduli space. This substack D(C0; P1) is the object I am mainly interested in. The main theorem 4.10 is stated in chapter 4. My result concerns the open substack D(C0; P1) of D(C0; P1), which relates to those stable curves with exactly 3g − 4 nodes, and describes it generically as a quotient stack. More importantly, this exhibits a dense open substack of D(C0; P1) as a quotient of a very simple moduli stack. There may be other ways to obtain this result. But on reading Edidin’s paper [Ed] I liked his approach, which made use of quotient stacks of Hilbert schemes, very much, and I decided that I wanted to apply this technique to my problem as well. Chapter 5 deals with a variation of this approach in order to include stable curves with 3g − 3 nodes as well. It turns out that one has to take care of each of the “extremal” curves separately. Let Dˆ (C0; P1) denote the one- point partial compactification of D(C0; P1) at the point representing C0, and let Dˆ(C0; P1) denote the corresponding substack of D(C0; P1). Then theorem 5.30, which describes the stack Dˆ(C0; P1) by a quotient stack, can be thought of as an equivariant version of the main theorem with respect to the action of the group of automorphisms of C0. The question of the compatibility and the glueing of the partial compact- ifications is discussed in chapter 6. This amounts mainly to the study of various groups of automorphisms associated to pointed stable curves. Finally, in chapter 7 all local information is collected to obtain some global results about the closed substack D(C0; P1). In particular, in proposition 7.10 a criterion is given to decide when the quotient description of D(C0; P1) extends to the partial compactification Dˆ(C0; P1). Essentially this is deter- mined by the group of automorphisms of C0.
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