Moduli space of curves and tautological relations

Transcription

1 MSc Mathematical Physics Master Thesis Moduli space of curves and tautological relations Author: Farrokh Labib Supervisor: prof. dr. Sergey Shadrin Examination date: July, 017 Korteweg-de Vries Institute for Mathematics

2 Abstract We start first with the construction of the moduli space of rational curves with marked points and quickly move on to curves of arbitrary genera with marked points. We then define the tautological ring and also the important ψ- and κ-classes. A set of additive generators is known for the tautological ring and the recent paper [PPZ16] by Pandharipande, Pixton and Zvonkine shows an algorithm to compute relations between these additive generators. We will look into this and compute several already known relations. Also attached to this thesis is a paper [KLLS17] written by R. Kramer, D. Lewanski, S. Shadrin and me where we exploit some polynomiality in the PPZrelations to give a new proof of Looijenga s result [Loo95] (RH g 1 (M g,1 ) = Q and RH >g 1 (M g,1 ) = 0) and also a more recent result by A. Buryak, S. Shadrin and D. Zvonkine [BSZ16] (RH g 1 (M g,n ) = Q n and RH >g 1 (M g,n ) = 0). Also we give a bound for the dimension of the tautological ring in lower degree. Title: Moduli space of curves and tautological relations Author: Farrokh Labib, farrokh.labib@student.uva.nl, Supervisor: prof. dr. Sergey Shadrin Second Examiner: dr. Hessel Posthuma Examination date: July, 017 Korteweg-de Vries Institute for Mathematics University of Amsterdam Science Park , 1098 XG Amsterdam

3 Contents Introduction 4 Acknowledgements 6 1 Moduli space of curves Universal families Moduli problems and moduli spaces Moduli space of smooth rational curves with n marked points Forgetful maps The boundary of M 0,n Moduli space of Riemann surfaces of genus g with n marked points Orbifolds M g,n as an orbifold Compactification of M g,n Cohomology of M g,n 34.1 The tautological ring ψ- and κ-classes Cohomological field theories Witten s r-spin class Givental s R-matrix action on CohFT s Action by translation Unit preserving action on CohFT s with unit Tautological relations Frobenius manifolds Outline of the program The topological field theory and the R-matrix The algorithm Explicit examples In M 0, In M 1, In M, Popular summary 6 3

4 Introduction Classifying certain objects up to a given notion of equivalence has been always an important goal in many areas of mathematics. One example is that of finite abelian groups: any finite abelian group is isomorphic to a finite direct sum of cyclic groups Z ki, i for some prime powers k i. We could call the set consisting of finite tuples of prime powers up to permutation the moduli space for classifying finite abelian groups. This means that for any point in this set, there is a unique isomorphism class of finite abelian groups given by the direct sum above and any finite abelian group corresponds to a point in this set. We see that this set is in bijection with isomorphism classes of finite abelian groups. But to be called a moduli space, we not only want a bijection with isomorphism classes of the objects at hand, but we also want this set to be a variety or scheme (or a more complicated topological space) and it should also have a universal property. These properties will be discussed thoroughly in chapter 1 of this thesis. Sometimes the space that parametrizes such objects are interesting topological spaces on their own. This happens to be the case when we want to classify Riemann surfaces of genus g with n marked points such that the automorphism group of this surface is finite. We will also say curve instead of Riemann surface. The equivalence here is an isomorphism of Riemann surfaces that respects the marked points. The moduli space will be denoted by M g,n and this space is not compact. There exists a compactification M g,n, the points that we add correspond to stable curves. A stable curve is a possibly reducible curve with possible nodes, such that the order of its automorphism group is finite. In section of chapter 1, we will see that this space admits a structure of a smooth complex orbifold of dimension 3g 3 + n. Since we are dealing with a very complicated space, the moduli space M g,n, one would like to compute topological invariants. One of the most interesting and useful invariants of topological spaces is cohomology. For g = 0 Keel gives a complete description in terms of generators and relations of the full cohomology ring H (M 0,n, Q) in [Kee9]. However, as of now there is still much to be discovered about the full cohomology ring of M g,n for higher genera. There is however a subring of H (M g,n ) which we call the tautological ring and denote by RH (M g,n, Q) for which we have a set of additive generators. This description will be given in chapter. The tautological ring is not freely generated by these generators, there are plenty of 4

5 relations between them. In a recent paper [PPZ16] by Pandharipande, Pixton and Zvonkine, they describe a way to obtain relations in the tautological ring. We will call such relations tautological relations. To explain how this algorithm of obtaining relations works, we need the concept of cohomological field theory, abbreviated by CohFT. One important example of a CohFT, and also the one which will be used to obtain relations, is Wittens r-spin class. It provides us for every r 3 a CohFT. There is Giventals R-matrix action on cohomological field theories which we will also need. These concepts are defined in chapter. Once we have these tools, we can describe how to obtain tautological relations using the paper by Pandharipande, Pixton and Zvonkine. We will call the relations that we obtain in this way PPZ-relations. The importance of the PPZ-relations is expressed in the conjecture that the PPZ-relations are all the relations in the tautological ring, i.e. any relation which holds in the tautological ring of M g,n can be found in the PPZrelations, possibly disguised. It is of course very hard to prove such a statement. We will see some explicit calculations and obtain well known relations in the tautological ring of M 0,4, M 1,1 and M,1. This is all done in chapter 3. Attached to this thesis is a paper [KLLS17] written by R. Kramer, D. Lewanski, S. Shadrin and me where we exploit some polynomiality in the PPZ-relations to give a new proof for the well known result by Looijenga: RH g 1 (M g,1 ) = Q and RH >g 1 (M g,1 ) = 0, see [Loo95]. We also give a new proof for a more recent result by S. Shadrin, A. Buryak and D. Zvonkine: RH g 1 (M g,n ) = Q n and RH >g 1 (M g,n ) = 0, see [BSZ16]. We also give a bound for the dimension of the tautological ring in lower degree. 5

6 Acknowledgenments I would like to take the opportunity here to thank my supervisor, Sergey Shadrin, for always being open to answer my questions, advising me during this project and involving me in his research group to work on very recent developments in this field. Farrokh Labib 6

7 1 Moduli space of curves In section 1.1 I will mostly use [KV07]. 1.1 Universal families Let us first start with a simple moduli problem. We want to classify all quadruples in P 1, that is, an ordered set of 4 points which are distinct. A first guess would be that the moduli space of this problem would be the variety Q = P 1 P 1 P 1 P 1 \ {diagonals} which indeed classifies all quadruples. But we want our moduli space to have more structure, not only that points of it are in bijection with the quadruples. We want it to carry a universal family. If Q carries a universal family, we will call it a fine moduli space. First what is a famliy of quadruples? Definition A family of quadruples in P 1 consists of a base variety B, a projection π : B P 1 B and four sections σ i, i = 1,..., 4 which are disjoint. We see that the fiber over a point b B is a copy of P 1 and the sections single out a quadruple in that fiber. So basically, a family of quadruples are quadruples parametrized by the base variety B. We could have also defined a famliy of quadruples in the following way: it is a morphism σ : B Q = P 1 P 1 P 1 P 1 \ {diagonals}. The disjointness of the sections is here captured in the fact that σ avoids the diagonals. Now we need the notion of pullback family. Definition 1.1. Suppose B P 1 B together with its sections is a family of quadruples. Let φ: B B be a morphism. Then B P 1 B is also a family where now the sections are the sections of the original family precomposed with φ and it is called the pullback family. Using the alternative definition of a family of quadruples, the pullback of the family B Q under φ: B B is just the composite B φ B Q. A family is then called universal if every other other family is induced from it by pullback. We are now ready to show that Q carries a universal family and hence is a fine moduli space for classifying quadruples in P 1. The sections are given by σ i ( p) = ( p, p i ), here p = (p 1, p, p 3, p 4 ) Q a quadruple. In the second viewpoint, the family is given by the identity map id Q : Q Q. We see that this family is tautological. From the second viewpoint it is then clear why this is a universal family: every morphism B Q factors through id Q : Q Q. The search for the fine moduli space which classifies quadruples in P 1 was not very 7

8 complicated. Let us now try to find the fine moduli space which classifies quadruples up to projective equivalence and this is where things get more interesting. Projective equivalence. The automorphism group of P 1 is isomorphic to PGL(). This is the group of matrices which are invertible modulo a constant factor. Then PGL() acts on P 1 by [ ] a b [x: y] = [ax + by : cx + dy]. c d Then two quadruples p and q are projectively equivalent if there is a φ PGL() such that φ(p i ) = q i. You can actually show that there is a unique automorphism of P 1 sending three distinct points p 1, p and p 3 to 0, 1 and respectively. In this way, we can define a map α: P 1 P 1 P 1 \ {diagonals} Aut(P 1 ). Now if we have a quadruple p, then define λ( p) P 1 \ {0, 1, } to be the image under the unique automorphism which sends the first three points to 0, 1,. This is called the cross ratio. The quadruple (0, 1,, λ( p)) is also called the normal form if the quadruple p. The cross ratio map λ: Q P 1 \ {0, 1, } is actually a morphism, since it is the composition of α id: Q Aut(P 1 ) P 1 and the map Aut(P 1 ) P 1 P 1 (defined by letting the first component act on the second component). It is now clear that any quadruple p is projectively equivalent to (0, 1,, λ( p)). By transitivity of the equivalence relation, we conclude that two quadruples are projectively equivalent if and only if their cross ratio is the same. This means that the set of equivalence classes of quadruples is in bijection with P 1 \ {0, 1, }. Denote this set by M 0,4. Here 4 corresponds to quadruples and 0 to the genus of P 1 which is 0. This notation will become clear later on. We can give this set structure of a variety because P 1 \ {0, 1, } is a variety. As mentioned before, we also want a universal family over M 0,4 so that we can say that it is a fine moduli space for classifying quadruples up to projective equivalence. The universal family over M 0,4. From now on I will identify M 0,4 with P 1 \{0, 1, }. We will give the description of the family from both viewpoints. In the first viewpoint we give the sections. The sections are σ 1 (q) = 0, σ (q) = 1, σ 3 (q) = and σ 4 (q) = (q, q) U : = M 0,4 P 1 for q M 0,4. In the second viewpoint, the morphism τ : M 0,4 Q is given by q (0, 1,, q). This family is now tautological in the sense τ λ that the composite M 0,4 Q M0,4 is the identity map. To actually show that this family has the universal property, we first need to define what equivalence of families is in this context. Projective equivalence in families. Let (B, {σ i } i ) and (B, {σ i } i) for i = 1,..., 4 be two families (with same base variety). Then these families are equivalent if there is 8

9 an automorphism φ: B P 1 B P 1 such that the following diagram commutes for each i = 1,..., 4 B P 1 B P 1 σ i B π φ id B After a moment of thought, we can say that in the second viewpoint, we want there to exist a morphism γ : B Aut(P 1 ) such that the following diagram commutes σ i B π B (γ,σ) Aut(P 1 ) Q B id B σ Q action Here σ : B Q and σ : B Q are two families. We will now prove that the tautological family τ : M 0,4 Q is a universal family. Lemma The tautological family τ : M 0,4 Q is a universal family. This means that for any other family σ : B Q there exists a unique morphism κ: B M 0,4 such that the pullback family denoted by κ τ is equivalent to the family σ. Proof So let σ be such an arbitrary family. We let κ = λ σ and show that this κ indeed works. The pullback of the family σ under κ is obtained by composition: B κ τ M 0,4 Q. This family does not need to be equal to the family σ, but we claim that it is projectively equivalent to σ. In total the pullback family is as follows B σ Q λ M 0,4 τ Q. The map Q λ τ M 0,4 Q sends a quadruple to its normal form. This can also be described by the map Q (α,id) Aut(P 1 ) Q action Q, hence we obtain a map B Aut(P 1 ) Q which is the desired map to show that the families are equivalent. We now only need to show that the morphism κ is actually unique. In order for κ τ to be equivalent to the family σ, we must have for a point b B that κ sends it to the cross-ratio of σ(b). This actually determines κ uniquely. Proposition M 0,4 is a fine moduli space for the problem of classifying quadruples up to projective equivalence in P 1. n-tuples in P 1. Let us now classify distinct n-tuples in P 1 up to projective equivalence. Let (p 1,..., p n ) be such n-tuple. Then by an automorphism of P 1 we can send the first three points to 0, 1 and. The last n 3 are then arbitrary, distinct and different from 0, 1 and. Hence the fine moduli space is M 0,n = M 0,4 M 0,4 \{diagonals}, where we take n 3 products of M 0,4. The universal family is given by π : M 0,n P 1 M 0,n together with the n sections which are given by: σ 1 ( p) = ( p, 0), σ ( p) = ( p, 1),σ 3 ( p) = 9

10 ( p, ) and σ i ( p) = ( p, p i ) where p = (0, 1,, p 3,..., p n ) M 0,n. What we have done here can also be formulated in a more general setting, namely we have solved the moduli problem of classifying all Riemann surfaces of genus 0 with n marked points up to isomorphism. This also explains the notation M 0,n. Before going in to this, we will discuss more generally what a moduli problem is and what a fine moduli space is for a moduli problem Moduli problems and moduli spaces In a moduli problem we want to classify certain geometric objects up to a given notion of equivalence. The geometric objects could be varieties or schemes, vector bundles, maps etc. of a specified type. The notion of equivalence depends in which category we are working. We then want to find a variety or scheme which is in natural bijective correspondence with the geometric objects up to equivalence. We will discuss what natural means here, but we saw it earlier in classifying quadruples up to projective equivalence. The notion of families and equivalence. We want our moduli space to capture and reflect the way the objects vary in families. So in all formulations of moduli problems, one begins with the notion of a family over a base scheme B and pullbacks of families over a morphism φ: B B. If X is a family over B, we denote it by X/B. Then the pullback along φ is a family over B and we denote it by φ X/B. Most of the times the family X/B is given by a map X B with extra structure. For example, for classifying quadruples the extra structure was the four disjoint sections. The pullback φ X/B family is often given by the usual fibre product: φ X := X B B and the second projection as the structure map. The induced extra structure on this morphism should also be made explicit. This pullback has the usual properties a pullback should have, namely pulling back along the identity morphism should give back the original family and pulling back along a composition φ ψ is the same as first pulling back along φ and then along ψ. There should also be a notion of equivalence of families compatible with the pullback operation. So if two families over B are equivalent, then their pullback along a morphism B B should also be equivalent as families over B. Fine moduli space. Now we come to the notion of a fine moduli space. Whenever you find a family U/M over a base scheme M with the property that if for any other family X/B there exists a unique morphism κ: B M with the property that κ U/B is equivalent to X/B, then we call U/M a universal family and the base scheme M is called a fine moduli space. Indeed, if M is a fine moduli space, then there is a bijection between families over B up to equivalence and morphisms B M. Also, the universal family is unique up to equivalence which is a consequence of its universal property. In the special case that B = {pt} is a point, we obtain the classification of objects. Namely, a family over a point is just an object and morphism {pt} M is a geometric point of M. So the geometric points of M are in bijective correspondence with the equivalence classes of families over a point, hence equivalence classes of objects. This is of course 10

11 one of the goals which a fine moduli space should solve and it is indeed solved using its universal property. Example Consider the moduli problem of classifying subsets. A family over a set B is defined to be a subset of B. Two subsets of B (i.e. two families over B) are considered equivalent when they are equal as subsets. If we have a subset A of B and a morphism φ: B B, then the pullback family is just φ 1 (A). Now it is clear that this problem admits a fine moduli space, namely M = {true, false} is the fine moduli space and {true} M is the universal family as you can check. Note that if you look at families over B = {pt}, the only families are B itself and the empty set. From what we have discussed earlier, families over a point are just the objects we classify. So basically, we classified the objects {pt} and using the moduli space M = {true, false} and it should be clear which geometric point of M corresponds to which object. Categorical reformulation. There is actually no fine moduli space for the moduli problem of classifying smooth curves of genus g 1 with n marked points such that g + n > 0 (this condition is for stability reasons) up to isomorphism in the category of schemes, but there is a so-called coarse moduli space M g,n. The best way of defining this notion is to use the language of category theory. Some basic knowledge of category theory is assumed here. The moduli functor. A moduli problem can be concisely encoded in a contravariant functor from the category of schemes Sch to the category of sets Set, F : Sch Set. This functor sends a scheme B to the set of equivalence classes of families over B. It sends a morphism φ: B B to the pullback map φ : F (B ) B as discussed above. This becomes a contraviant functor by the axioms imposed on pullback. So it respects compositions of morphisms and the identity morphism. Also the compatibility of the pullback with equivalence classes of families makes this functor well-defined. Representable functors. Given any scheme M, we can construct the following contravariant functor h M : Sch Set, by sending a scheme B to the set of morphisms from B to M, that is Hom(B, M). It is called the functor of points of M. A morphism φ: B B is sent to the morphism Hom(B, M) Hom(B, M) by sending β to β φ where β Hom(B, M). Then, if a functor F is isomorphic to a representable functor h M by a natural transformation η : h M F, it is called representable and we say that the pair (η, M) represents F. We can replace η by an element in F (M) using Yoneda s lemma, say U which is a representative of an equivalence class of families over M. Now we are ready for the categorical reformulation. Namely, Lemma A family U/M is a universal family (so M is a fine moduli space) for F if and only if the pair (U, M) represents F. 11

12 Proof Suppose U/M is a universal family. We need to find a natural isomorphism η : h M F. It is natural to define η B (κ) = κ U where κ: B M. Since U/M is a universal family, we see that η B : Hom(B, M) F (B) is a bijection. We still need to show the naturality of η, meaning the following diagram should commute, Hom(B, M) h M (φ) η B F (B) F (φ) η B Hom(B, M) F (B ) where φ: B B is a morphism. Commutativity of this diagram follows immediately using properties of the pullback, namely (κ φ) U = φ (κ U). We conclude that η is a natural isomorphism between F and h M. By Yoneda s lemma, the element in F (M) that corresponds to h M is h M (id M ) = U. Hence the pair (U, M) represents F. To prove the converse, we just need to show that the family h M (id M ) =: U is the universal family over M that we are looking for. But this is immediate using the naturality of the isomorphism between h M and F. Coarse moduli space. For many moduli problems there does not exist a fine moduli space. We want the moduli space of such moduli problems to still have that the geometric points are in bijection with equivalence classes of the objects, but we will relax the universal property it should have. We say that the pair (M, ν) is a coarse moduli space for a moduli functor F, where M is a scheme and ν : F h M is a natural transformation, if the following holds: (1). The universal property here is that (M, ν) is initial among all such pairs. This means that if (M, ν ) is another such pair, then there exists a unique morphism ψ : M M such that ν = ψ ν. Actually, by Yoneda s lemma, morphisms M M correspond bijectively to natural transformations h M h M. In other words, any natural transformation ν : F h M factors uniquely through ν. In this sense h M is the representable functor closest to F. (). The set map ν {pt} : F ({pt}) Hom({pt}, M) is a bijection. This means that the geometric points of M are in bijection with the equivalence classes of objects. Example Let us reformulate the problem of the previous example using the moduli functor. The moduli functor F in this problem sends B to the set of all subsets of B. By the lemma above, this functor F should be naturally isomorphic to the functor h M which sends B to the set of morphisms of B to M = {true, false}. The natural transformation between these functor is as follows. η B : F (B) h M (B) sends a subset of B to the morphism which maps this subset on the point {true} and the rest on the point {false}. One can check that this transformation is indeed natural and also an isomorphism between F and h M. 1

13 1.1. Moduli space of smooth rational curves with n marked points A projective smooth rational curve is a projective smooth curve of genus 0. Here we are looking at objects (C, x 1,..., x n ) where C is a projective smooth rational curve and x 1,..., x n are n 3 distinct marked points on C. An isomorphism between two such objects (C, x 1,..., x n ) and (C, x 1,..., x n) is first of all an isomorphism φ: C C with the extra condition that φ(x i ) = x i for i = 1,..., n. An automorphism of (C, x 1,..., x n ) is an isomorphism φ: C C which preserves the marked points. Since we have n 3, there are no non-trivial automorphisms. We then say that π : X B with n disjoint sections σ i : B X is a family of n-pointed rational curves if π is flat and proper and the geometric fibers X b := π 1 (b) is a projective smooth rational curve. Then n sections single out n marked points in each fiber. An isomorphism between two families π : X B and π : X B over the same base is an isomorphism φ: X X such that the following diagram commutes. σ i X X B φ π σ i π Now let us compare families of smooth rational n-pointed curves with n-tuples of points in a fixed P 1, i.e. the case we did earlier. One can actually show that if π : X B admits at least one section, then it is isomorphic to P(E) where E is some rank vector bundle over B ([Har13] page 369). Also, if there are at least two disjoint sections, then the bundle splits. If there are at least 3 disjoint sections, one can actually show that X = B P 1. So the families we discussed earlier are actually the same as we just defined. Hence Proposition For n 3, there is a fine moduli space M 0,n solving the moduli problem of classifying smooth projective rational curves with n marked points up to isomorphism. Indeed, this is just a reformulation of what we have done earlier. The universal family is just the trivial family U 0,n := M 0,n P 1 M 0,n. Now, the space M 0,4 is not compact hence M 0,n is not compact. In general, compact spaces are much nicer than non-compact ones, so we really want to look for a compactification. First we compactify M 0,4 and deduce from that how to compactify M 0,n. We saw that M 0,4 = P 1 \ {0, 1, }. A first idea could be to just let the marked points coincide. We will show now that with this compactification, geometric information will be lost. Consider the following two families of marked curves C t = (P 1, 0, 1,, t) and D t := (P 1, 0, 1/t,, 1). Since the cross-ratios of the marked points are the same, these two families of marked curves are isomorphic. Now let us see what happens if t 0. If we look at C 0, we see that the first marked point (which is 0) coincides with the 4th marked point. If we look at D 0, we see that the nd marked point coincides with the 3rd marked point. So we see that in the limit t 0 the two families of marked curves D t and C t degenerate in different objects, i.e. B 13

14 they are not projectively equivalent. Since both viewpoints are correct (we just look at the curve in two different coordinate systems), the correct way to compactify is to add the following stable curve So what happens here is that whenever two marked points tend to each other, there sprouts out a new bubble with these marked points on it. To be more precise, let us look at the family of curves given by the equation xy = t (or xy = tz in homogeneous coordinates). On these curves, mark the following points: [x 1 : y 1 : z 1 ] = [0 : 1 : 0], [x : y : z ] = [1 : 0 : 0], [x 3 : y 3 : z 3 ] = [1 : t : 1], [x 4 : y 4 : z 4 ] = [t : 1 : 1]. The graph looks as follows for t 0 Now as t 0 we see that the curve degenerates in to two P 1 s, which we call bubbles, with on each bubble two marked points. Inspired by this example we get to the following definition. Definition A stable curve of genus 0, also called a rational stable curve, with n marked points is a connected curve with irreducible components, which we will also call twigs, isomorphic to P 1, the irreducible components intersect transversally and the arithmetic genus of the dual graph representing the stable curve is 0. Due to stability reasons, every irreducible component should have at least 3 marked points, here points of intersection with other components, which we will call nodes, are also considered as marked points. Furthermore, marked points are not allowed to be on a node. 14

15 An explanation is required here. Let us look at an example. You should have the following pictures in mind when thinking about stable curves of genus 0 with n marked points To each such stable curve, we associate a dual graph. Each irreducible component is given by a vertex, the vertex has a label 0 which corresponds to the genus of the component. If two components intersect transversally, we draw an edge between the two vertices. Marked points are given by leaves/legs. So the dual graphs associated to the previous example is Now we give non-examples of genus 0 stable curves. The first has arithmetic genus 1, the second has a marked point which is on a node so it is not allowed and the last one is not stable because it has not enough marked points on the first bubble. An automorphism of a stable n-pointed curve is an automorphism on each twig such that it keeps the special points fixed. Since on each twig there are at least three special points, we conclude that stable curves are automorphism-free, meaning that the identity 15

16 is the only automorphism. Then, a family of stable n-pointed curves is a flat and proper map π : X B together with n disjoint sections, which are also disjoint from the nodes, such that each geometric fiber X b := π 1 (b) is a stable n-pointed curve. Isomorphisms of two such families is the same notion as that for smooth curves. Theorem (Knudsen [Knu83]). For each n 3, there exists a fine moduli space M 0,n for classifying stable n-pointed curves. This space is a smooth projective variety and it contains M 0,n as a dense open subset. Example Let us look at the universal family U 0,4 M 0,4. We already know that U 0,4 = M 0,4 M 0,4 \ {diagonal}. If we just put back the missing points in M 0,4 and close up the diagonal, we will get in trouble with the sections; they will not be disjoint anymore at the points corresponding to the stable curves we added to M 0,4. Namely, the sections are σ 1 (x) = (0, x), σ (x) = (1, x), σ 3 (x) = (, x) and the diagonal section σ 4 (x) = (x, x). The diagonal section will intersect the other sections in the bad points (0, 0), (1, 1) and (, ). So instead of taking P 1 P 1 as the universal curve, we blow it up at the bad points (0, 0), (1, 1) and (, ) and define U 0,4 = Bl(P 1 P 1 ). Denote by E 0, E 1 and E the exceptional divisor. You can think of them as P 1 s attached to the three bad points. The sections σ i transform and we obtain a commutative diagram Bl(P 1 P 1 ) P 1 P 1 σ i σ i M 0,4 The transformed sections σ i will no longer be disjoint, since the divisors determined by the diagonal section σ 4 intersects transversally the divisors determined by σ i for i = 1,, 3, hence in the transform of the sections, they determine different points on the exceptional divisors. So the fiber over the three added points in M 0,4, namely 0, 1 and, which correspond to stable curves, is exactly that stable curve Forgetful maps In this subsection we will start constructing M 0,n recursively. Adding a marked point. Suppose (C, x 1,..., x n ) is a stable curve and given is an arbitrary point q on the curve C. We want to construct a stable curve (C, x 1,..., x n, q) from these data. If q is not a special point on the curve, i.e. not one of the marked points or a node, then (C, x 1,..., x n, q) is obviously a stable n + 1-pointed curve. Otherwise, we have two different cases which can be solved canonically. (1) If q coincides with one of the marked points, say x i for some i, then create a new bubble and place the points x i and q on it. It does not matter where on this P 1 we place these two points as long as they are distinct, since there are three special points. The bubble intersects C transversally on the point where x i used to be. 16

17 () When q coincides with one of the nodes, create a new bubble in between with q on it. Again it does not matter where q is placed, since there are three special points. For example, q q This process of adding a mark is compatible with families as the next proposition shows. Proposition (Knudsen [Knu83]) Let (X /B, σ 1,..., σ n ) be a family of stable n- pointed curves and let δ : B X be an extra section. Then there exists a family (X /B, σ 1,..., σ n+1 ) of stable n + 1-pointed curves and a B-morphism φ: X X such that (1) the restriction φ 1 (X \ δ) X \ δ is an isomorphism, () φ σ n+1 = δ, (3) φ σ i = σ i for i = 1,..., n. This family is unique up to isomorphism. Furthermore, this process of adding marks commutes with fiber products. Forgetting a mark. We want to construct a morphism π : M 0,n+1 M 0,n by forgetting the last marked point. Note however that it does not matter which point we forget, forgetting the last marked point is for notational convenience. Let us now describe the forgetful map on sets. Suppose we have a stable curve (C, x 1,..., x n+1 ) and we want to forget the last marked point. It is possible that the curve (C, x 1,..., x n ) is not stable anymore, because there could be a vertex with less than three leaves going out (an edge is also counted in this case). In that case, we stabilize the curve by contracting the component of the curve which is not stable anymore. We illustrate it with an example. If the curve is given by the dual graph x 11 0 x

18 and we forget the last mark x 11, then the corresponding component of the curve will not be stable anymore since it has only two special points. We then contract it and obtain the following stable curve x so the marked point x 10 will be placed on position of the node. This process of forgetting marks is also compatible with families. Proposition (Knudsen [Knu83]) Let (X /B, σ 1,..., σ n ) be a family of stable n + 1-pointed curves. Then there exists a family (X /B, σ 1,..., σ n) of stable n-pointed curves together with a B-morphism φ: X X such that (1) φ σ i = σ i for i = 1, dots, n, () for each b B, the induced morphism X b X b twig and it contracts an eventual unstable twig. is an isomorphism on each stable This family is unique up to isomorphism and we say that it is the family obtained from X /B by forgetting σ n+1. Furthermore, forgetting sections commutes with fiber products. We described the forgetful map π : M 0,n+1 M 0,n on sets, but together with this proposition we can show that it is actually a morphism of schemes. Indeed, let us look at the universal family U 0,n+1 M n+1. By forgetting the last section, we obtain a family of n-pointed curves with base scheme M 0,n+1, hence by the universal property of M 0,n, we obtain a unique morphism π : M 0,n+1 M 0,n. It is clear that this map coincides with π which we only defined on sets. Hence π is indeed a morphism. Sketch of construction M 0,n. To construct M 0,n as a fine moduli space, we show that U 0,n = M0,n+1 and do the construction iteratively. So {pt} = M 0,3 U 0,3 = M0,4 U 0,4 = M0,5... We will show now that M 0,5 = U 0,4 and construct the universal family U 0,5. This works similarly for general n 5. We want a bijection (as sets) between M 0,5 and U 0,4. Let q U 0,4 and consider the universal family π : U 0,4 M 0,4. Define F q := π 1 (π(q)), which is the fiber passing through q, this means that F q represents the point π(q) M 0,4 so it is a 4-pointed stable curve. The point q is another point on the stable curve F q and hence we obtain a 5-pointed curve C q = (F q, q) by adding a marked point as discussed 18

19 above. We see that this map q C q M 0,5 is injective. On the other hand, given a 5-pointed stable curve (C, x 1,..., x 5 ), we obtain a stable 4-pointed curve by forgetting x 5. The place where x 5 was together with the 4-pointed curve specifies a fiber of π together with a point in it, which is a point in U 0,4. This proves the bijection. Let us now construct the universal family U 0,5 for M 0,5 to conclude that it is a fine moduli space. As in the case of constructing U 0,4, we consider the fiber product U 0,4 M0,4 U 0,4, add a diagonal section and then stabilize. U 0,5 φ U 0,4 M0,4 U 0,4 U 0,4 δ π U 0,4 M 0,4 This fiber product together with the sections induced by σ i : M 0,4 U 0,4 is a family of 4-pointed curves which is parametrized by U 0,4. There is actually another natural section δ which is the the diagonal section. Since this section will not be disjoint from the 4 other sections, we need a stabilization to obtain a family. By proposition we can actually do this and we denote it by U 0,5 and φ is the map from the proposition. Now we want to show that the fiber (U 0,5 ) q over a point q U 0,4 is indeed a 5-pointed curve isomorphic to C q where q C q is the bijection between U 0,4 and M 0,5. The fiber over q in U 0,4 M0,4 U 0,4 is the curve F q = π 1 (π(q)), a stable 4-pointed curve. The diagonal section singles out one more special point which is q itself, hence we have a not necessarily stable 5-pointed curve (F q, q). We want the fiber of the stabilization U 0,5, but by proposition stabilization commutes with fiber products, i.e. the fiber of the stabilization is the stabilization of the fiber. The stabilization of (F q, q) is exactly C q as we have discussed when we constructed the bijection between M 0,5 and U 0,4. Hence we have shown that (U 0,5 ) q = Cq. To conclude that M 0,5 is a fine moduli space, we still need to check that this family is universal, this is left to the reader. Remark Keel shows in [Kee9] how to construct U 0,n explicitly using a sequence of blow-ups. We have seen the example U 0,4 earlier. Also Kapranov in [Kap93] constructs U 0,n using a sequence of blow-ups, but he uses another sequence The boundary of M 0,n We have seen that we have to add the dual graphs with edges to compactify M 0,n. Definition The closure of a configuration given by a dual graph is a smooth and irreducible subvariety of M 0,n and is called a boundary stratum. Note that for a given dual graph, the boundary stratum corresponding to this graph consist also of curves with further degenerations. 19

20 Now we can define a canonical isomorphism between products of moduli spaces and boundary strata. Let us do this for boundary strata where the dual graph has only one edge, such boundary strata are also called boundary divisors. Denote by A for set of marked points on one vertex and B for the set of marked points on the other vertex. So A B is a partition of the marked points. Note that by stability A, B. Let us denote such divisors by D(A B). The first vertex with marked points A parametrizes curves with marks A together with one more special point, namely the node. Call this extra mark x. So the moduli space for this vertex is M 0,A {x}. The same holds for the other vertex, it parametrizes curves with marks B and x, so the moduli space there is M 0,B {x}. So we have an attaching map M 0,A {x} M 0,B {x} D(A B). This is actually an isomorphism. Since M 0,n is smooth and irreducible, so are products of moduli spaces, hence D(A B) is a smooth and irreducible variety. Since the dimension of M 0,k is k 3, we can count the dimension of D(A B) by counting the dimension of M 0,A {x} M 0,B {x} which is A B = A + B 4 = n 4. We have now done the dimension computation for boundary divisors, but it is now clear how to proceed and prove that for any other boundary stratum for which the dual graph has δ nodes, the dimension will be n 3 δ. Example To get a better understanding of boundary strata, let us find intersections of boundary divisors in M 0,6. Denote by a, b, c, d, e and f the marks on the curves. We want, for example, to find the intersection of the subvarieties D(a, b c, d, e, f) and D(a, b, c d, e, f). As mentioned before, a boundary stratum given by a dual graph, contains all further degenerations. A general point in D(a, b c, d, e, f) is of the form b a c 0 0 but we can further degenerate to obtain stable curves of the form f d e b a c f d e Hence D(a, b c, d, e, f) D(a, b, c d, e, f) = D(a, b c d, e, f). Example In the previous example, we saw that D(a, b c, d, e, f) D(a, b, c d, e, f) was not empty. The reason why it is not empty is because {a, b} {a, b, c} and this implies 0

21 that after a degeneration of such curves, with c going to the node, both divisors contain such stable curves. In general, if A B and A B are partitions of the marked points and there are no inclusions among the subsets A, B, A, B, then the intersection D(A B) D(A B ) will be empty. Relations between boundary divisors. For general n, we can find relations in the Chow ring of M 0,n between divisors by pulling back divisors of M 0,4 along forgetful morphisms. Let us look more precisely into this. Let π : M 0,n+1 M 0,n be the forgetful morphism forgetting the last marked point, say x. Let A B be a partition of the n marked points. Then the inverse image of D(A B) under π is simply D(A {x} B) D(A b {x}), since before forgetting x, it was either on the component with marks given by A or it was on the component with marks given by B. Since the geometric fibers are reduced, we conclude that in the Chow ring of M 0,n+1 π (D(A B)) = D(A {x} B) + D(A b {x}). We know that in P 1 all points are rationally equivalent. So D(a, b c, d) D(a, c b, d) D(a, d b, c) in the Chow ring of M 0,4. If we pull this divisor back using π : M 0,5 M 0,4, we obtain the following relation CH(M 0,5 ). D(a, b, x c, d)+d(a, b c, d, x) D(a, c, x b, d)+d(a, c b, d, x) D(a, d, x b, c)+d(a, d b, c, x). We can go on and pull this back many more times and obtain the following relation in CH(M 0,n ). D(A B) D(A B) D(A B). (1.1) a,b A c,d B a,c A b,d B a,d A b,c B Keel actually shows in [Kee9] that the Chow ring of M 0,n is generated by boundary divisors and the relations 1.1 together with the relations given in the previous example are all the relations they satisfy. 1. Moduli space of Riemann surfaces of genus g with n marked points In the previous section we discussed in detail M 0,n. Now we want to look at the moduli space of curves of genus g with marked points. We are going to do this in a different category since in the realm of schemes there is no fine moduli space parametrizing such curves. The existence of non-trivial automorphism seems to be the obstruction to the existence of a fine moduli space as scheme. If we still want to find a fine moduli space, we need to enlarge the category we are working in. If we want to work completely algebraicgeometrically, we need the notion of algebraic stacks. If we want to work analytically, 1

22 we need the notion of an orbifold. Here the existence of automorphism is encoded in the point of the orbifold itself as the stabilizer of that point. Since the theory of algebraic stacks is more complicated, we will follow the easier route of orbifolds. In the next subsection we will start defining complex orbifolds Orbifolds In this subsection we will define basic notions in the theory of orbifolds. I am following the paper [CR01]. Definition 1..1 Let U be a connected topological space, V an n-dimensional smooth connected manifold and let G be a finite group G which acts on V smoothly. We say that (V, G, π) is a uniformizing system if π : V U is a continuous map inducing a homeomorphism between V/G and U. Let (V, G, π) and (V, G, π ) be two uniformizing systems for U. They are isomorphic when there exists (1) a group isomorphism λ: G G, () a diffeomorphism φ: V V that is λ-equivariant, (3) and we have that π = π φ. Suppose we have an automorphism (φ, λ) of a uniformizing system (V, G, π). Denote by ker(g) the subgroup of G which acts trivially on all of V. Since π φ = π, we see that φ maps orbits to orbits and hence φ(x) = g(x) x for some g : V G. But φ should be smooth, hence the map g must be smooth, but G has the discrete topology, hence φ(x) = g x for some fixed g G. We can also say something about λ, namely λ(h) = ghg 1 where g is the same. This can be shown as follows: φ is λ-equivariant, so g (h x) = φ(h x) = λ(h)φ(x) = λ(h) (g x) so we see that for all x V (λ(h)g) x = (gh) x, but then λ(h)g = ghg 0 g 1 for some g 0 ker(g). Since λ is a group automorphism it follows that g 0 = e hence λ(h) = ghg 1. Definition 1.. Let U and (V, G, π) be as above. Let ι: U U be a connected open subset of U and let (V, G, π ) be a uniformizing system for U. We then say that (V, G, π ) is induced from (V, G, π) if (1) we have a monomorphism λ: G G, () we have a λ-equivariant open embedding φ: V V, (3) λ is an isomorphism from ker(g ) to ker(g), (4) ι π = π φ.

23 We also call (φ, λ): (V, G, π ) (V, G, π) an injection. Two injections (φ i, λ i ): (V i, G i, π i ) (V, G, π) (i = 1, ) are isomorphic when there is an isomorphism (ψ, τ): (V 1, G 1, π 1 ) (V, G, π ) and an automorphism (ψ, τ) of (V, G, π) such that (ψ, τ) (φ 1, λ 1 ) = (φ, λ ) (ψ, τ). Lemma 1..3 Let (V, G, π) be uniformizing system for U. Let U U be a connected open subset. Then (V, G, π) induces a unique isomorphism class of uniformizing systems for U. Proof Existence: The preimage of U under π consists of connected components in V and G acts on this set of connected components by permutation. Consider then the subgroup G G that fixes a chosen connected component V. Then (V, G, π = π V ) becomes a uniformizing system for U. Uniqueness: Choosing different connected component of π 1 (U ) yields isomorphic uniformizing system. Suppose now (Ṽ, G, π) is any other induced uniformizing system for U. We want to show that (V, G, π ) and (Ṽ, G, π) are isomorphic. Let (ψ, τ) be the injection of (Ṽ, G, π) into (V, G, π). We will show that (ψ, τ) induces the sought for isomorphism. Suppose ψ(ṽ ) lies in V (if not, choose another connected component and the uniformizing systems are isomorphic). We will show now that ψ(ṽ ) V is closed. Let {x n } n N and such that y 0 := lim n ψ(x n ) exists. Since π is surjective, there exists z 0 Ṽ such that π(z 0) = π(y 0 ). Also by surjectivity, there should exist {z n } n N Ṽ converging to z 0 such that π(z n ) = π(ψ(x n )). But by commutativity of the following diagram V U ψ Ṽ π V π we see that π(z n ) = π(x n ). Then there should exist {a n } n N G such that a n z n = x n since they should be in the same G-orbit. But G is finite, hence a n = a becomes constant for n > N and N large enough. We then see that x n a z 0 and y 0 = ψ(a z 0 ). In other words, ψ(ṽ ) V is closed. Since ψ is also an open embedding, we see that ψ is a diffeomorphism between Ṽ and V. Now ψ is τ-equivariant, so τ( G) G. We also have that τ gives an isomorphism between ker( G) and ker(g ) and we can conclude now that (Ṽ, G, π) is isomorphic to (V, G, π ). Let U be connected and locally connected topological space. We will assume this from now on implicitly. Now let us define what a germ of uniformizing systems is at p U. Suppose we have two uniformizing systems (V 1, G 1, π 1 ) and (V, G, π ) for two neighborhoods U 1 and U respectively. Then we say that (V 1, G 1, π 1 ) and (V, G, π ) are isomorphic at p/define the same germ at p if they induce isomorphic uniformizing systems for a neighborhood U 3 of p. We are now ready to define what an orbifold is. Definition 1..4 Let X be a Hausdorff, second countable topological space. An n- dimensional orbifold structure on X is giving by the data: for each p X there is a 3

24 neighborhood U p of p together with a uniformizing system (V p, G p, π p ) such that for any point q U p the uniformizing systems (V p, G p, π p ) and (V q, G q, π q ) define the same germ at q. The germ of orbifold structures is defined as follows: two orbifold structures, which we denote by {(V p, G p, π p ): p X} and {(V p, G p, π p): p X}, are equivalent if for any p X the uniformizing systems (V p, G p, π p ) and (V p, G p, π p) are isomorphic at p, i.e. define the same germ at p. Given a germ of orbifold structure on X we say that X is an orbifold. We call each (V p, G p, π p ) a chart at p and U p a uniformized neighborhood. An open set U X is uniformized if there exist a uniformization (V, G, π) of U such that for any p U the uniformizations (V, G, π) and (V p, G p, π p ) define the same germ at p. Definition 1..5 Let X be an orbifold. Let p X and (V p, G p, π p ) a chart. Then p is called regular or smooth if G p is trivial. If G p is not trivial, p is called singular. We denote by X reg the set of smooth points and by ΣX the set of singular points. We will call X reduced if G p acts effectively. We can see that every orbifold X induces a reduced orbifold X red. We do this by redefining the local group. We will call X red the reduced associate to X. Example (Global quotients). Let M be a smooth n-manifold and G a finite group acting on it smoothly. Let us put an obifold structure on X := M/G. A global uniformizing system is given by (M, G, π) where π : M M/G is the natural projection map. For any point x X, choose a connected open neighborhood U x of x. Choose a connected component of π 1 (U x ), say Ũx, and let G x be the subgroup of G such that G x Ũx Ũx. Then (Ũx, G x, π Ũx ) is a uniformizing system for U x. Example We are going to give S the structure of an orbifold. Let D s and D n be neighborhoods of the south and north pole respectively such that S = D s D n. Uniformize D s and D n by (D s, Z, π s ) and (D n, Z 3, π n ) respectively. Here the groups act by rotation. To show that this makes S an orbifold, we need to show that if p D s D n that (D s, Z, π s ) and (D n, Z 3, π n ) define the same germ at p, but this is easily seen to be the case. This orbifold structure can not be given by a global quotient, since D s D n induces non-isomorphic uniformizing systems when viewed as an open subset of D s or of D n. Morphism of orbifolds. Now we want to define morphisms between orbifolds. Let U and U be uniformized topological spaces which are uniformized by the systems (V, G, π) and (V, G, π ) and let f : U U be a continuous map. A C k lifting of f, for 0 k, is a C k map f : V V such that the diagram f V V π π f U U 4

25 commutes and for any g G there exists g G such that for all x V we have f(g x) = g f(x). We will also denote such a lifting as f : (V, G, π) (V, G, π ). Suppose we have two liftings f i : (V i, G i, π i ) (V i, G i, π i ) (i = 1, ). We say that f 1 and f are isomorphic if there exists isomorphisms of uniformizing systems (φ, τ): (V 1, G 1, π 1 ) (V, G, π ) and (φ, τ ): (V 1, G 1, π 1 ) (V, G, π ) such that the diagram f 1 V 1 V 1 φ φ f V V commutes. Let p U. Consider uniformized neighborhoods U p of p and U f(p) of f(p) such that f(u p ) U f(p). If f is a lifting of f, then this will induce a lifting f p for the continuous map f Up : U p U f(p) as follows: let U p be uniformized by (V p, G p, π p ) and (φ, τ): (V p, G p, π p ) (V, G, π) be any injection, the choice of injection is irrelevant since by lemma 1..3 they are all isomorphic. So for simplicity, just assume it is given by a connected component of π 1 (U p ). Consider now the map f φ: V p V. We see immediately that π f φ(v p ) U f(p). Therefore f φ(v p ) lies in a connected component of (π ) 1 (U f(p) ). This connected component gives rise to an injection (φ, τ ): (V f(p), G f(p), π f(p) ) (V, G, π ) such that f φ(v p ) φ (V f(p) ). Then define f p := (φ ) 1 f φ: V p V f(p). This is then the sought for lifting f p : (V p, G p, π p ) (V f(p), G f(p), π f(p) ) of f Up : U p U f(p). When we make a different choice of an injection (φ, τ): (V p, G p, π p ) (V, G, π), it will give an isomorphic lifting as one can check. The germ of liftings is defined as follows: two liftings are equivalent at p if they induce isomorphic liftings on a smaller neighborhood of p. Let us now consider orbifolds X and X together with a continuous map f : X X on topological spaces. A lifting of f is then defined as follows: let p X and (V p, G p, π p ) a chart around p and (V f(p), G f(p), π f(p) ) a chart around f(p), then we want a lifting f p of f πp(v p) : π p (V p ) π f(p) (V f(p) ) such that for any q π p (V p ) we have that f p and f q define the same germ of liftings of f at q. The germ of liftings is defined as follows: two liftings of f denoted by { f p,i : p X} for i = 1, are equivalent if for each p X, fp,1 and f p, define the same germ of liftings of f at p. Definition 1..6 With the notation as above, we say that a C k map between X and X is a germ of C k liftings of f which we denote by f. Example Let X = R C. We put a global orbifold structure on X by the chart (R C, Z 4, π). Here Z 4 acts on C by multiplication by 1. Define C 1 maps f i : R (R C, Z 4, π) by f 1 (t) = (t, t ) and f (t) = (t, t ) when t 0 and f (t) = (t, 1t ) when t > 0. These two maps induce the same map on topological spaces. Let us see if they are isomorphic as C 1 maps between orbifolds. We need to find an automorphism (φ, τ): (R C, Z 4, π) (R C, Z 4, π) such that φ f 1 = f, but it is clear that such a φ does not exist: such a map φ must satisfy φ((t, t )) = (t, t ) for t 0 and φ((t, t )) = 5

26 (t, 1t ) for t > 0 so that φ f 1 = f, but this φ does not satisfy φ(x) = g x for a fixed g Z 4 for all x R C. Hence f 1 and f are not equivalent C 1 maps. Orbifold bundle. We now look at the notion of orbifold bundles which in the theory of manifolds corresponds to smooth vector bundles. Instead of saying orbifold vector bundle, we will sometimes also say orbibundle. Let U be a uniformized topological space and (V, G, π) a uniformizing system for U. Let E be a topological space together with a continuous surjective map p: E U. Then a uniformizing system of rank k orbifold bundle for E consists of (1) a uniformizing system (V R k, G, π) for E. Here the action on V R k is given by g (x, v) = (g x, ρ(x, g)v) and ρ: V G Aut(R k ) is a smooth map such that ρ(g x, h) ρ(x, g) = ρ(x, hg) for all g, h G and x V. () The natural projection map p: V R k V satisfies π p = p π, i.e. the following diagram commutes. V R k E π p p V U π Isomorphism between uniformizing systems of an orbifold bundle E is defined similarly as in the usual case except that the diffeomorphism V R k V R k maps fibers of p: V R k V to the same fiber and is linear. When U U is a connected open subset, we can also show that this induces a unique isomorphism class of uniformizing systems of orbifold bundle for E = p 1 (U ). The germ of uniformizing systems of orbifold bundle at x U is also defined in a similar way. Definition 1..7 Let X be an orbifold and E a topological space together with a surjective continuous map p: E X. We then say that E has the structure of a rank k orbifold bundle over X if for each x X there is a uniformized neighborhood U x and a uniformizing system of rank k orbifold bundle for E x := p 1 (U x ) over U x such that for any other y U x, the uniformizing systems of orbifold bundle over U x and U y define the same germ at y. Given two structures of rank k orbifold bundle over X which we concisely denote by {p 1 (U x ): x X} and {p 1 (Ũx): x X}, we then say that they define the same germ of rank k orbifold bundle over X if for each x X the uniformizing systems of orbifold bundle for p 1 (U x ) and p 1 (Ũx) define the same germ at x. The topological space E together with a given germ of orbifold bundle structures becomes an orbifold and it is called an orbifold bundle over X. The charts (V x R k, G x, π x ) are called local trivializations of E. For each x X we have that E x = p 1 (x) is isomorphic to R k /G x. Two orbifold bundles over X, p 1 : E 1 X and p : E X, are isomorphic if there exists a smooth map ψ : E 1 E given by ψ x : (V 1,x R k, G 1,x, π 1,x ) (V,x R k, G,x, π,x ) which induces an isomorphism between (V 1,x, G 1,x, π 1,x ) and (V,x, G,x, π,x ) and it is a linear isomorphism between the fibers of p 1 and p. Complex orbifold bundles or rank k over X are defined by replacing R with C. 6

27 The next thing we want to do is define vector fields, differential forms etc. on orbifolds. These notions can be defined as smooth sections of some orbifold bundles, just as in the case of manifolds where vector fields are sections of the tangent bundle and differential forms are sections of the exterior products of the cotangent bundle. Definition 1..8 Let X be an orbifold and p: E X a rank k orbifold bundle. A C l map s: X E is called a C l section if locally s is given by s x : V x V x R k, s x is G x -equivariant and p s x = id Vx. Orbifold bundles can also be described using transition maps (see [Sat57]) just as it can be used to define vector bundles. We can construct an orbifold bundle using the following data: a compatible cover U of X such that for any injection i: (V, G, π ) (V, G, π) we have a smooth map g i : V Aut(R k ) which together gives an open embedding V R k V R k given by (x, v) (i(x), g i (x)v). We must also have for any composition j i: (V, G, π ) (V, G, π ) (V, G, π) that g j i (x) = g j (i(x)) g i (x), for all x V. (1.) Two collections of such maps g 1 and g define isomorphic orbifold bundles if there are maps δ V : V Aut(R k ) such that for any injection i: (V, G, π ) (V, G, π) we have g i (x) = δ V (i(x)) g 1 i (x) (δ V (x)) 1, for all x V. Example We can construct the tangent bundle T X for an orbifold X by defining g i : V Aut(R k ) to be the Jacobian matrix. It is non-singular since i: (V, G, π ) (V, G, π) is an injection. It clearly satisfies 1. by the chain rule. In the same way, we can define the cotanget bundle T X by letting the transition functions gi be the dual of the inverse of the transition functions for the tangent bundle, so gi (x) = (g i(x) 1 ) t. More constructions. 1. behaves naturally under operations of vector spaces such as tensor products, exterior product etc. hence we can define vector fields, tensor fields, differential forms etc. as C sections of the corresponding bundles. There is also a de Rham cohomology theory for orbifolds which is isomorphic to the singular cohomology of the underlying topological space X. Characteristic classes. One can also construct Chern classes using Chern-Weil theory for orbifold bundles. General orbifolds may not have any local sections, so the construction of connections and curvature of connections will not work for such orbifolds. We say that an orbifold vector bundle is good if the local group of the base and total space have the same kernel. It is known that good orbibundles always have enough local sections. Examples of good orbifold vector bundles are the tangent and cotangent bundles together with all their tensor products and exterior products. Throughout this part, we assume that our orbifold vector bundles are good. Let U be uniformized by (V, G, π) and p: E U a rank k orbibundle which is uniformized by (V R k, G, π). We want to define a connection on this orbibundle. For this, consider G-equivariant connections on V R k, i.e. for any smooth section u and vector 7

28 field v on V and any g G we want g( v u) = gv gu. Now, let (V i R k, G i, π i ) (i = 1, ) be two isomorphic uniformizing systems of E and suppose we have G i -equivariant connections i on each of them. Then 1 is isomorphic to if there is an isomorphism ψ : (V 1 R k, G 1, π 1 ) (V R k, G, π ) such that ψ 1 =. One can also check that for a connected open subset U U, an isomorphism class of connections over U induces a unique isomorphism class of connections over U. We then say that 1 and are equivalent at p, i.e. define the same germ, if they induce isomorphic connections over a neighborhood of p. Definition 1..9 Let p: E X be an orbibundle with orbibundle structure given by B = {(V p R k, G p, π p )}. A collection of connections { p : p X}, with each p being G p -equivariant, defines a connection on E if for any q U p = π p (V p ) we have p is equivalent to q at q. If i (i = 1, ) are two connections in the reference of orbibundle structures B i (i = 1, ), then they are equivalent if for any p X, they induce isomorphic connections on a neighborhood of p. Suppose we have two connections 1 and on p: E X in the reference to orbibundle structures B 1 and B. By this we mean that the two structures are isomorphic, but 1 is given by a collection of connections on {(V 1p R k, G 1p, π 1p )} and is given by a collection of connections on {(V p R k, G p, π p )} where B 1 = {(V 1p R k, G 1p, π 1p ): p X} and B = {(V p R k, G p, π p ): p X}. Then for each p X there is a uniformized U p such that B 1 and B induce isomorphic uniformizing systems of p 1 (U p ) which we denote by (V p R k, G p, π p ). Then 1 and induce G p -equivariant connections 1p and p on (V p R k, G p, π p ). Their difference is then a G p -equivariant, End(R k )-valued smooth 1-form on V p. Thus it follows that the difference of two connections is a C section of the orbibundle T X End(E). On the other hand, if we add a C section of the orbibundle T X End(E) to a connection, we obtain yet another connection on E. Proposition The set of all connections of the orbibundle E X is a non-empty affine space modelled on the C sections of the orbibundle T X End(E). Now we want to define curvature F ( ) for a connection on E X. Suppose is defined in reference to the orbibundle structure {(V p R k, G p, π) p : p X}. Then the curvature of p is a G p -equivariant smooth -form with values in End(R k ). The collection of these curvatures define a C section of Λ (T X) End(E) and we denote it by F ( ). The Chern-Weil construction applied to each F ( p ) yield characteristic classes for orbibundles. Proposition To each invariant polynomial, there is a characteristic class: to each orbibundle E X there is an associated cohomology class in the de Rham cohomology (or singular cohomology) of X which only depends on the isomorphism class of the orbibundle. As a consequence we have Chern classes, Pontrjagin classes and Euler class in the category of orbibundles. Example Let (Σ, z 1,..., z k ) be smooth complex curve of genus g Σ together with set of marked points denoten by z 1,..., z k. We can give this curve the structure of orbifold as 8

29 follows. Choose m i 1 for i = 1,..., k and at each z i denote by D i a disk neighborhood of z i. A uniformizing system for D i is then the branched covering D i D i given by z z m i. Denote by L Σ the cotangent orbifold bundle. The first Chern number is given by c 1 (L Σ )([Σ]) = g Σ + k i=1 (1 1mi ). This shows that characteristic classes are in general rational classes. This is one of the reasons why we will consider singular cohomology of an orbifold with rational coefficients. Pullback orbibundle. Suppose p: E X is an orbibundle and f : X X is a C map between orbifolds. A pullback orbibundle of E over X via f is given by an orbibundle π : E X together with a C map f : E E such that each local lifting of f is an isomorphism when restricted to each fiber and f covers the map f between the bases. 1.. M g,n as an orbifold The moduli space of Riemann surfaces of genus g with n marked points (such that g + n > 0) can be given a structure of a smooth complex orbifold of dimension 3g 3 + n. It turns out that it will be a fine moduli space as an orbifold. Theorem 1..1 ([HM06], C) Let C be a Riemann surface of genus g with n marked points such that g + n > 0. Let G be its group of automorphisms, it is finite by the condition g + n > 0. There exists (1) an open simply connected domain U C 3g 3+n, () a family p: U U of Riemann surfaces of genus g with n marked points, (3) a group G acting on U which commutes with p and thus the action descends to an action on U and it satisfies the conditions: (i) the fiber C 0 over 0 is isomorphic to C, (ii) G becomes the automorphism group of C 0, (iii) For any other family of smooth curves X B such that C b is isomorphic to C for some C, then there is a subset b B B such that the family restricted to B is a pullback of the family U U by a morphism B U. With this theorem, we construct two smooth complex orbifolds M g,n and U g,n. Using the charts U/G we can construct M g,n, but we will not go into the details. Also, using the opens sets U we construct an orbifold U g,n and there is an orbifold map p: U g,n M g,n. This is the universal family which makes M g,n a fine moduli space for classifying smooth Riemann surfaces of genus g with n marked points. It is important to note that the automorphism group of a curve given by t M g,n is exactly the stabilizer of t. 9

30 Example Let us find M 1,1. Riemann surfaces of genus 1 are also called elliptic curves. Any elliptic curve can be given by the quotient of C by a rank lattice. The image of 0 C becomes naturally a marked point. We conclude M 1,1 = {lattices}/c. If we want a more explicit description, we need to know a little bit more about the correspondence between elliptic curves and lattices. Let L 1 and L be two lattices. Then the elliptic curves C/L 1 and C/L are isomorphic if and only if the lattices L 1 and L are homothetic. Two lattices L 1 and L are homothetic if there exist α C such that L 1 = αl. Let (z 1, z ) be a basis for a lattice L. Then we can immediately divide by z to obtain a different basis (τ, 1), but we see that L τ is of course homothetic to L. Moreover, we can take τ H in the upper half plane. Lemma Let L τ and L τ be lattices given by the bases (τ, 1) and (τ, 1). Then L τ is homothetic to L τ if and only of there exist γ SL(, Z) such that γ τ = τ. Here the action is given by [ ] a b γ τ = τ = aτ + b c d cτ + d. Proof Suppose αl τ = L τ. Since αl τ L τ, we may find a, b, c, d Z such that ατ α = aτ + b = cτ + d hence τ = aτ + b cτ + d. Using the other inclusion, we find for some a, b, c, d Z Hence the matrices [ a ] b c d τ = a τ + b c τ + d. and [ a b ] c d are invertible and they are each others inverse. Hence they are in SL(, Z). Suppose now that there exists γ SL(, Z) such that γ τ = τ. Then define α := cτ + d and we see that ατ = aτ + b and we have shown that αl τ L τ. For the other inclusion, use γ 1. We conclude that M 1,1 = H/SL(, Z). The elements {id, id} act trivially on lattices, hence we can actually take the group PSL(, Z) = SL(, Z)/{id, id} to be our symmetry group. The fundamental domain of the group PSL(, Z) is shown below and one sees explicitly how M 1,1 looks like by identifying the arcs AB and AB and the half-lines BC and B C. 30

31 Given a lattice L given by (τ, 1) where τ is in the fundamental domain, the stabilizer is a subgroup in SL(, Z) and it is the automorphism group of C/L which is in accordance with the theorem. Let us find what the stabilizer is of the following lattices, in other words automorphism of the following elliptic curves In the generic case (case A) there is only one symmetry, besides the identity, namely central symmetry. So the stabilizer is Z. In the case where the lattice is given by (i, 1), we can rotate by π/ and the group of automorphisms is Z 4. This elliptic curve corresponds to the point A in the fundamental domain. In the last case, where the lattice is given by ( 1+i 3, 1) we can rotate by π/3 and the symmetry group becomes Z 6. This lattice corresponds to the point B (or B ) in the fundamental domain Compactification of M g,n The compactification of M g,n is similar to as what we have done to compactify M 0,n. To compactify, we allow stable curves, which are curves with singularities. The only type of singularity which is allowed is a simple node, i.e. locally it is given by the equation xy = 0. Definition A stable curve of genus g with n marked points is a compact complex algebraic curve (C, x 1,..., x n ) such that 31