Moduli Spaces of Pointed Rational Curves

Transcription

1 Moduli Saces of Pointed Rational Curves Renzo Cavalieri Lecture Notes Graduate Student School Combinatorial Algebraic Geometry rogram at the Fields Institute July 8-, 06

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3 Contents First Moduli Saces: M 0,n 7. A few generalities on moduli saces Moduli Problems Moduli Functors Universal Families Moduli of n Points on P Comactification: first stes Further exercises Stable Pointed Curves: M 0,n 9. The boundary Natural morhisms Forgetful morhisms and universal families Gluing morhisms Intersection Theory on M 0,n A quick and dirty introduction to intersection theory. 7.. Chern Classes of Bundles The Chow ring of M 0,n Psi classes Smoothing nodes and the normal bundle to a boundary divisor Proerties of si classes Further exercises Weighted Pointed Curves 9. Stability and Hasset saces Relation to Karanov s Construction Psi classes on Hasset saces Losev-Manin saces Further exercises

4 Troical M 0,n 9. Dirty introduction to troical geometry Troical M 0,n Troical Weighted stable curves

5 Introduction

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7 Lecture First Moduli Saces: M 0,n. A few generalities on moduli saces.. Moduli Problems In order to ose a moduli roblem we need the following ingredients:. A class of (geometric) objects P in some category C.. A notion of family of such objects.. The notion of equivalence of families. Families over one oint are recisely the objects of P, and hence we are in articular assigning the notion of equivalence of objects. Intuition. A moduli sace M P for the above roblem consists of a sace in the same category as the objects arameterized, such that: oints are in bijective corresondence with equivalence classes of objects in P. there is a natural bijection between families over a given base B and functions B M P. We now define what a family is, in the familiar category of toological saces. We leave it as an exercise to generalize the definition to any case we may be interested in. Definition. Let P define a class of objects in the category C of toological saces. Then for any B C, a family of P-objects over B is a toological sace X together with a surjective continuous function π X B such that the fiber at each oint (X b = π (b) for b B) is an object in P. 7

8 8 Exercise. Formulate the notion of a family in the language of categories. Note that you need to require your category C to have fiber roducts. Also, the role of the oint is layed by a terminal object in C. If we have a sace M P whose oints are in bijection with objects in P, then a family naturally defines a set function ϕ π B M P : ϕ π (b) = [X b ] (.) The image of a oint b is the oint in the moduli sace is the (equivalence class of the) fiber of that oint. In order to call M P a moduli sace we like the most (a fine moduli sace) we make the following requirements:. That the set function ϕ π is continuous (a morhism in C).. That no two non-equivalent families give the same function.. That every continuous function from f B M P arises as the function associated to a family. When the three conditions above are verified, there a fruitful dictionary between the geometry of the moduli sace and the geometry of families of objects. Becoming fluent with using this dictionary and translating questions back and forth is one of the main goals of this mini-course. Intuition. If this whole dictionary with families seems a bit weird, let me try to convince you that in fact it is quite natural. We would like the geometry of the moduli sace to reflect the similarity of objects. For examle, if our objects had (a comlete set of) metric invariants, then we would like the moduli sace to also have a metric, and objects with very close invariants corresond to oints that are very near each other in the moduli sace. Or, even better, that moving the invariants continuously would result in a continuous ath in the moduli sace. Of course the roblem is making such statements general and recise. The notion of a family achieves recisely that: it tells us that over a base B objects are varying in an accetable way if they fit together to form a larger object still in the right category. Exercise. Consider the moduli roblem of isomorhism classes of unit length lane segments u to rigid motion in the lane. Note that if such a sace exists it can have only one oint. Construct two families of segments over a circle that are not equivalent to each other. But they both must give the constant function: hence fails... Moduli Functors We now reformulate the discussion in our revious section in categorical language. For the uroses of this mini-course this level of abstraction is hardly necessary, but it is useful to be able to connect to this tye of language.

9 9 Definition. A moduli roblem (or moduli functor) for a class P of objects in a category C with fiber roducts is a contravariant functor: defined by: F P C Sets for an object B C, F P (B) is the set of isomorhism classes of families of P-objects. for a morhism f B B, F P (f) is the set ma sending a family X B to the ullback family f (X) B. Definition. An object M P C reresents the functor F P if the functor of oints (Hom C (, M P )) of M P is isomorhic (as a functor) to the moduli functor via a natural transformation T. If such an object exsits, it is called a fine moduli sace for the moduli roblem. Exercise. Meditate for a few minutes until you convince yourself that this is equivalent to the above discussion about families and the natural bijection with functions to the moduli sace. In articular, note that evaluating the two functors on the terminal object in your category recovers the invoked bijection between oints of the moduli sace and (equivalence classes of) objects you want to arameterize. Intuition. If you are funked out by this idea of understanding a sace by thinking of its functor of oints, think of it as a generalization of what we do when we talk about abstract manifolds. We give u the idea of understanding the manifold via a set of global coordinates, and rather focus on the local data. A differentiable atlas is the notion of understanding a collection of (injective) functions that cover the set of oints of the manifold (the charts), lus the transition functions on double overlas. Here instead we want to understand ALL ossible functions into M P, and all ossible ullbacks. This is of course not a roof (the formal roof is given by Yoneda s lemma), but it should make it fairly lausible that the knowledge of the functor of oints should be enough to recover the (scheme, manifold, etc) structure of the moduli sace... Universal Families When a moduli functor is reresentable by a fine moduli sace, there is a very secial object with a ma to the moduli sace, such that the fiber over each oint m M P is recisely the (an) object (in the equivalence class) arameterized by the oint m. Definition. Given a fine moduli sace M P, let T denote the natural transformation identifying the functor of oints of M P with the moduli functor. The universal Family π U P M P

10 0 is defined to be T( MP ). The universal family has the roerty that for any m M P corresonding to an (equivalence class of) object(s) [X m ] P, [π (m)] = [X m ] (.) Exercise. A nice consequence of the existence of a universal family is that any other family of P-objects is obtained from the universal family by ullback. Show that this is just a formal consequence of the categorical definitions. Exercise. Any scheme X is a fine moduli sace for the functor families of oints of X, the universal family being X itself. This is all a big tautology, but make sure it makes erfect sense to you. It is a good exercise to unravel the definitions and kee them straight.. Moduli of n Points on P We now introduce the moduli sace M 0,n of isomorhism classes of n ordered distinct marked oints i P. The subscrit 0 denotes the genus of the curve P.Since this is our first meaningful class of moduli saces, we sell out all the ingredients. We work in the category of algebraic varieties (or schemes if you refer) over C. The class of objects we consider is P = {(,..., n ) i P, i /= j for i /= j}. (.) The equivalence relation on objects is (,..., n ) (q,..., q n ) if there is Φ Aut(P ) s.t., for all i, Φ( i ) = q i. (.) A family of n distinct oints on P over a base B consists of the following data: a (flat and roer) ma π X B, such that, for every oint b B, X b = π (b) P. n disjoint sections σ i B X. A function s B X is called a section of π X B if π s = B. This means that for every b B, s(b) X b, i.e. s icks exactly one oint in every fiber of the family. Two sections are called disjoint if their images are disjoint. Two families (X, Bπ, σ,..., σ n ) and ( X, B, π, σ,..., σ n ) are equivalent if there exists an isomorhism Φ X X making the following diagram

11 commute: X Φ X (.) σ σ n Finally we introduce our moduli functor. Definition. The moduli functor π B π σ σ n M 0,n Schemes Sets assigns to any scheme B the set of equivalence classes of families of n distinct oints on P, and to any morhism f B B the set ma induced by the ullback of families. Exercise 6. Show that the definitions of family of n distinct oints on P and of equivalence of families secialise to the definitions of objects and equivalence of objects when B = t. Exercise 7. For n show that the set M 0,n (t.) consists of only one oint. Show that for some base sace B, there exist non-equivalent families. Conclude that a single oint sace does not reresent the functor M 0,n. For n =, since the automorhism grou Aut(P ) = PGL (C) allows us to move any three oints on P to the ordered trile (0,, ) (see Exercise ), the set M 0, (t.) consists of a single oint. Given any family (X, B, π, σ, σ, σ ), consider the ma T X B P defined by: T (x) = (π(x), CR Xπ(x) (σ (π(x)), σ (π(x)), σ (π(x)), x)), where CR Xπ(x) is the cross-ratio (as defined in Exercise ) on the fiber containing the oint x (we know such fiber is isomorhic to P and that the cross-ratio is indeendent of the choice of isomorhism one may choose to define it, hence T is well defined). T realizes an isomorhism of families between (X, B, π, σ, σ, σ ) and the family (B P, B, π, 0,, ), where π is rojection on the first factor and the sections are constant sections. Exercise 8. Show that the discussion in the revious aragrah roves that the functor M 0, is reresented by the algebraic variety M 0, = t. Show that a universal family is given by U = (P, t., π, 0,, ). Intuition. The fact that any trile of oints can be moved into (0,, ) via an automorhism means that there is one equivalence class of triles, and so, if the functor is reresentable, it has to be reresented by a one oint

12 sace. The fact that there is exactly one automorhism moving a trile to the distinguished trile we chose to reresent our equivalence class causes the fact that all families of triles can be trivialized. Equivalently, one can say that the only automorhism of P that fixes (0,, ) is the identity. As a general hilosohy, when we arameterize objects which have no non-trivial automorhisms, we are not able to glue locally trivial families in non trivial ways. I.e., non-trivial automorhism of the objects we arameterize tend to give obstructions to the reresentability of the functor. Going one ste u, the functor M 0, is reresented by the variety M 0, = P {0,, } : given a quadrule (,,, ), we can always erform the unique automorhism of P sending (,, ) to (0,, ); the isomorhism class of the quadrule is then determined by the image of the fourth oint. A universal family U is given by (M 0, P, t, π, 0,,, δ), where δ is the diagonal section δ() =. This family is reresented in Figure.. (, ) (,) U 0 (0,0) IP 0 M 0, Figure.: The universal family U M 0,. Exercise 9. By choosing to trivialize the first three sections, we have made a choice of a global coordinate system on M 0,, or alternatively, we gave an injective ma from P {0,, } to the set of configurations of four oints on

13 P that meets each equivalence class exactly once. By trivializing a different choice of sections we make a different choice of coordinates, and corresondingly get a different universal family. Show exlicitly (gotta do it once in your life) one change of coordinates for M 0,, and write down the induced isomorhism of universal families. The general case is similar. Any n-tule = (,..., n ) is equivalent to a n-tule of the form (0,,, Φ cr ( ),..., Φ cr ( n )), where Φ cr is the unique automorhism of P sending (,, ) to (0,, ). This shows M 0,n = n times M 0,... M 0, {all diagonals}. If we define U n = M 0,n P, then the rojection of U n onto the first factor gives rise to a universal family U n π σ i M 0,n where the σ i s are the universal sections: σ i () = (, Φ cr ( i )) U n. (.6) This family is tautological since the fibre over a moduli oint, which is the class of a marked curve, is the marked curve itself. With U n as its universal family, the affine variety M 0,n is a fine moduli sace for isomorhism classes of n ordered distinct marked oints on P.. Comactification: first stes. We now understand quite well the moduli sace M 0,n. In articular, we notice it is not comact for n. There are many reasons why comactness (or roerness) is an extremely desirable roerty for moduli saces. As an extremely ractical reason, roer (and if ossible rojective) varieties are better suited for an intersection theory, which then allows for using our moduli saces for enumerative geometric alications. Also, a comact moduli sace encodes information on how objects can degenerate in families. A totally reasonable, if a bit naively hrased, question like what haens when in M 0,? would find an answer if we find a good comactification for M 0,. In general there are many ways to comactify a sace. A good comactification M of a moduli sace M should have the following roerties:

14 . M should be itself a moduli sace, arametrizing some natural generalization of the objects of M.. M should not be a horribly singular sace.. the boundary M M should be a normal crossing divisor.. it should be ossible to describe boundary strata combinatorially in terms of simler objects. This oint may aear mysterious, but it will be clarified soon enough. We discuss the simle examle of M 0,, to rovide intuition for the ideas and techniques used to comactify the moduli saces of n-ointed rational curves for arbitrary n. A natural first attemt is to just allow the oints to come together, i.e. enlarge the collection of objects P that we are considering from P with ordered distinct marked oints to P with ordered, not necessarily distinct, marked oints. After all, as a set, P = (P ), which is retty nice. When we imose our equivalence relation, we notice two kinds of issues: We have equivalence classes corresonding to all oints, three oints, or two airs of oints having come together. Such equivalence classes behave like one or two-ointed curves; as we saw in Exercise 7 there exist nontrivial families giving rise to constant mas, which revents reresentability of our functor. We have six equivalence classes corresonding to where exactly two oints have come together. These oints look like M 0, s, so they may be allright. But consider the families over B = C with coordinate t: C t = (0,,, t) and D t = (0, t,, ). For each t /= 0, the automorhism of P φ t (z) = z t shows that C t D t, thus corresonding to the same family in M 0,. But for t = 0, C 0 has = whereas D 0 has =. These configurations are not equivalent u to an automorhism of P, hence are distinct oints in our comactification of M 0,. Thus, we have a family with two distinct limit oints (the sace is nonsearated). Our failed attemt was not comletely worthless though since it allowed us to understand that:. we should not allow too many oints to coincide; in articular, we should always have at least three distinct marks on P.. we want the condition = to coincide with =, and likewise for the other two ossible disjoint airs.

15 On the one hand this is very romising: is the number of oints needed to comactify P {0,, } to P. On the other hand, it is now mysterious what modular interretation to give to this comactification. The strategy is to achieve the modular interretation for the comactification by comactifying simultaneoulsy the moduli sace and the universal family. Let us then turn to the universal family, illustrated in Figure., and naively comactify the icture by filling in the three oints on the base, and comleting U to P P where the sections are extended by continuity. Intuition. We notice a bothersome asymmetry in this icture: the oint is the only one allowed to come together with all the other oints: yet common sense, backed u by the exlicit examle just resented, suggests that there should be democracy among the four oints. One way to restore democracy is to blow-u P P at the three oints (0, 0), (, ), (, ). This makes all the sections disjoint, and still reserve the smoothness and rojectivity of our universal family. Exercise 0. Refer back to Exercise 9, consider the isomorhism ψ of the two different universal families U, U obtained when different choices of sections to be trivialized are made. Note that if the universal families are closed to U (res. U ) P P, ψ extends to a rational function that is indeterminate at the oints (0, 0), (, ), (, ), and contracts the fibers over 0,,. Show that blowing u such three oints in both U and U resolves the indeterminacies of ψ, which now extends to an isomorhism ˆψ Bl(U ) Bl(U ). Exercise 0 shows that the candidate for a universal family Bl(U ) is well behaved with resect to the change of coordinates induced by different choices of three sections to trivialize. The fibres over the three excetional oints are P E i : nodal rational curves. These are the new objects that we have to allow in order to obtain a good comactification of M 0,. This leads us to the notion of stable rational ointed curve, which we will exand uon in the next lecture.. Further exercises Exercise (Automorhisms of P ). Deending on your favorite oint of view, we are arameterizing configurations of marked oints on the comlex rojective line P (algebraic geometry) or on the Riemann shere Ŝ = C (comlex analysis). We are interested in the grou of automorhisms of P, that can be described as: PGL(, C): equivalence classes of matrices with non-zero determinant, u to simultaneous non-zero scaling of all the coefficients, acting on oints of P by matrix multilication on the homogeneous coordinates.

16 6 Moebius tranformations: Meromorhic functions of the form: with ad bc /= 0 w = az + b cz + d, If you are not familiar with these concets, send some time understanding each of them and their equivalence. Show that there is a unique automorhism of P that mas any three distinct oints of P to any other three distinct oints. A slick way to do so is to show that for any (,, ), with i P and i /= j when i /= j, there exists a unique automorshism Φ cr Aut(P ) such that Φ cr ( ) = 0, Φ cr ( ) =, Φ cr ( ) =. (.7) Exercise (Cross-ratio). Given (,,, ), distinct oints of P, their cross-ratio is defined to be: With Φ cr as in (.7), show: CR(,,, ) = ( )( ) ( )( ) CR(,,, ) = Φ c r( ). (.8) Show that the cross ratio is a rojective invariant: i.e., if Φ is any automorhism of P CR(,,, ) = CR(Φ( ), Φ( ), Φ( ), Φ( )). Exercise (A new ersective on a encil of conics). Consider the encil of lane conics defined by the equation: S = {λx(z y) + µz(y x) = 0} P P (.9). Recognize that the total sace of this encil is the blow-u of P at four oints. For each i =,..., denote E i the excetional divisor corresonding to blowing u the i-th oint.. Consider the commutative diagram: U = M 0, P F P P, (.0) where F is the function: π M 0, λ (λ ) π P F (λ, w) = (w(w ) w(λ ) λ(w )). (.)

17 7 (a) Show that the Zariski closure of Im(F ) in P P is S. (b) Show that for each i =,...,, F σ i gives a arameterization of the excetional divisor E i. Thus we have realized the comactified universal family π U M 0, as the blow-u of P at oints, and identified the four tautological sections with the excetional divisors. Exercise. Notice that M 0, is equal to the universal family U 0,, where the images of the three distinguished sections are the three oints added to comactify M 0,. Do you think this is a coincidence, or is there a reason for this?

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19 Lecture Stable Pointed Curves: M 0,n At the end of our last lecture we learned that in order to comactify M 0, to a moduli sace, we had to invite to the moduli roblem some degenerations of the rojective line: secifically, airs of rojective lines glued together at one oint. In fact allowing rational curves with nodal singularities is a way to comactify M 0,n for all n. We now make this recise. Definition 6. A stable rational n-ointed curve is a tule (C,,..., n ), such that: C is a connected curve of arithmetic genus 0, whose only singularities are nodes (i.e. locally analytically around a singular oint Sing(C), C can be defined by the equation xy = 0.) (,..., n ) are distinct oints of C Sing(C). The only automorhism of C that reserves the marked oints is C. For more combinatorially minded eole, this is equivalent to the notion of a stable tree of rojective lines. Definition 7. A stable n ointed tree of rojective lines is a tule (C,,..., n ), such that: C is connected and each irreducible comonent of C is isomorhic to P. Irriducible comonents are called twigs. The oints of intersection of the comonents are ordinary double oints. There are no closed circuits in C, i.e., if any node is removed then the curve becomes disconnected.,..., n and the double oints of C are all distinct and are called secial oints. Every twig has at least three secial oints. 9

20 0 Figure.: Stable marked trees of rojective lines. The oints should be labeled, but I got lazy... We draw a marked tree as in Figure., where each line reresents a twig. Exercise. Show that Definitions 6 and 7 are equivalent. Exercise 6. Develo the natural definition of family of rational stable n- ointed curves over a base B and of isomorhism of families; describe the moduli functor M 0,n for isomorhism classes of such families. Intuition. Here s an intuitive (albeit imrecise) icture about the comactification to stable ointed curves. When oints on a rational curve C want to coincide, we don t allow them as follows: at the moment of collision, we create a new bubble attached to the oint of collision and transfer the oints to the new bubble. At this oint we are able to act via elements of PGL(, C) on each comonent of the curve, but such elements must be comatible in the sense that they must reserve nodes. That is why nodes of C are considered just like marked oints. With all these definitions in lace, in Section. we roved the following statement. Lemma. M 0, P reresents the functor of isomorhism classes of families of n-ointed rational curves. The universal family U may be viewed as the second rojection Bl (0,0),(,),(, ) (P P ) P, or as the blow u of a encil of lane conics at its four base oints. This result generalizes to arbitrary n.

21 Theorem (Knudsen). There exists an irreducible, smooth, rojective fine moduli sace M 0,n for n-ointed rational stable curves, comactifying M 0,n. The universal family U n is obtained from U n via a finite sequence of blow-us. We discuss the roof of this theorem and a few constructions of M 0,n in Exercise 8. For references in the literature, you may see [KV07] or [Knu8], [Knu], [Kee9] or [Ka9b]. At this moment we turn our attention to exloring the geometry of this family of moduli saces.. The boundary We define the boundary to be the comlement of M 0,n in M 0,n. It consists of all oints arameterizing nodal stable curves. We begin by making some urely set-theoretical considerations. The boundary of M 0, consists of three oints arameterizing the nodal ointed curves deicted in Figure.. The boundary of M 0, is reresented in Figure.: it consists of oints arameterizing ointed curves with three irreducible comonents, and 0 coies of M 0, arameterizing rational ointed curves with two irreducible comonents. In general the boundary of M 0,n is stratified by locally closed subsets arameterizing curves of a given toological tye, together with a rescribed assignment of the marks to the irreducible comonents of the curve. A natural way to encode this data is via the dual grah to a ointed curve. Definition 8. Given a rational, stable n-ointed curve (C,,..., n ), its dual grah is defined to have: a vertex for each irreducible comonent of C; an edge for each node of C, joining the aroriate vertices; a labeled half edge for each mark, emanating from the aoriate vertex. Figure.: The boundary of M 0, consists of three oints.

22 boundary strata codimension boundary strata codimension Figure.: Boundary strata of M 0,

23 Figure.: Two boundary strata of M 0, on the left, and their dual grahs on theright. Figure. gives an examle of the dual grahs of some strata in M 0,. Now we can say that a boundary stratum S is identified by the dual grah Γ S of the general curve that it arameterizes. Exercise 7. Show that the codimension of a boundary stratum S equals the number of nodes that any curve arameterized by S has, or, equivalently, the numberof edges in its dual grah. The closures of the codimension boundary strata of M 0,n are called the irreducible boundary divisors; they are in one-to-one corresondence with all ways of artitioning [n] = A A c with the cardinality of both A and A c strictly greater than. We denote D(A) = D(A c ) the divisor corresonding to the artition A, A c. Exercise 8. Describe the set theoretic intersection of two divisors D(A) D(A ). In articular describe combinatorial conditions for A and A that characterize when the intersection is emty. There is a natural artial order on the strata: S < S if the grah Γ S can be obtained from Γ S by contracting some edges. Geometrically, this means that S is in the closure of S. Exercise 9. Describe exlicitly the oset of boundary strata for M 0,. Show that each codimension-two stratum is in the closure of exactly two codimension-one strata.

24 . Natural morhisms There is a wealth of natural morhisms among the various moduli saces M 0,n. To be as intuitive as ossible, we now describe them as set functions, i.e. by describing how they act on the objects arameterized. There is a fair amount of technical work shoved under the rug that shows that all of these constructions work in families and are therefore honest morhisms of the moduli saces. Intuition. Understanding these morhisms is a owerful tool in the study of the geometry of these moduli saces: it shows that the boundary of M 0,n is built out of smaller moduli saces of ointed rational curves, and it oens the way to an inductive study of any geometric question about M 0,n that can be ushed to the boundary... Forgetful morhisms and universal families The (n + )-th forgetful morhism is the function that assignes to an (n + ) ointed curve, the n ointed curve obtained by removing the last marked oint: π n+ M 0,n+ M 0,n (.) (C,,..., n+ ) (C,,..., n ) The function π n+ is defined if n+ does not belong to a twig with only three secial oints, as the n-ointed curve obtained by forgetting n+ is still stable. If it does belong to such a twig, then the resulting tree is no longer stable. One obtains a stable curve again by contracting the unstable twigs, as illustrated in Figure.. Intuition. Generically, the data of an (n + ) ointed curve can be thought as of the choice of a oint - namely n+ - on an n ointed curve. Intuitively this means that one should be able to identify the universal family of M 0,n with M 0,n+. This is indeed the case, and it is at the core of Knudsen s construction of the moduli sace in [Knu8, Knu]. The contraction morhism assigns to (C,,..., n+ ): c M 0,n+ U 0,n (.) n+ (C,,..., n ) if (C,,..., n ) is stable; k π n+ (C,,..., n+ ), if n+ is on a twig with three secial oints, one of which is k ;

25 n+ i I contraction Text i In+ k contraction k Figure.: Contracting twigs that become unstable after forgetting the n+. In the second case the mark k is laced where the node used to be. n π n+ (C,,..., n+ ), if n+ is on a twig with three secial oints, the other two being nodes; n is the node which is formed after contracting the twig that contained n+. The stabilization morhism is defined by: s U 0,n M 0,n+ (.) s[ (C,,..., n )] = (C,,..., n, ) if is a smooth oint of C and it is not a mark; s[ k (C,,..., n )] is the curve defined by attaching a twig at the osition of the i-th mark, and utting k and n+ on this twig, as in Figure.6; if n is a node of C, then s[n k (C,,..., n )] is the curve defined by inserting a twig between the two shadows of the node n and lacing n+ on this twig, as in Figure.7. Theorem ([Knu8, Knu]). The mas c, s are well defined morhisms, they are inverses of each other, and they make the following diagram com-

26 6 stabilization I n+ i k I n+ i k Figure.6: The image of a mark under the stabilization morhism stabilization n n+ Figure.7: The image of a node under the stabilization morhism

27 7 mute: U 0,n s c M 0,n+ π π n+ M 0,n This identifies (M 0,n+, π n+ ) with the universal family of M 0,n. Exercise 0. Describe the image of s σ i in M 0,n+. Remark. There is nothing secial about forgetting the last mark. We have forgetful morhisms π i that forget any mark we want (and therefore different ways of viewing M 0,n+ as a universal family over M 0,n ). Also, we may comose forgetful morhisms to forget more than one oint at a time.. Gluing morhisms There are also natural gluing morhisms; for [n] = I I c a artition of the set of indices, with I and I c, gl I M 0,I M 0,I c M 0,n takes a air of ointed curves and identifies the oints marked and. The resulting nodal curve is clearly stable. The ma gl I is an isomorhism onto its image, which is the boundary divisor corresonding to the artition [n] = I I c. This shows that the closure of the codimension one boundary strata are the irreducible comonents of the boundary. One can realize the closure of any boundary stratum as the isomorhic image of an aroriate gluing morhism. This statement makes recise the earlier claim that the boundary of M 0,n is built out" of roducts of smaller saces of rational ointed stable curves.. Intersection Theory on M 0,n.. A quick and dirty introduction to intersection theory The main character here is the Chow ring, A (X), of a smooth algebraic variety X. The ring A (X) is, in some loose sense, the algebraic counterart of the cohomology ring H (X), and it makes recise in the algebraic category the intuitive concets of oriented intersection in toology. Elements of the grou A n (X) are formal finite sums of codimension n closed subvarieties (cycles), modulo an equivalence relation called rational equivalence. A (X) = dimx 0 A n (X) is a graded ring with roduct given by intersection.

28 8 Intersection is indeendent of the choice of reresentatives for the equivalence classes. In toology, if we are interested in the cu roduct of two cohomology classes a and b, we can choose reresentatives a and b that are transverse to each other. We can assume this since transversality is a generic condition: if a and b are not transverse then we can erturb them ever so slightly and make them transverse while not changing their classes. This being the case, then a b reresents the cu roduct class a b. In algebraic geometry, even though this idea must remain the backbone of our intuition, things are a bit trickier. Excetional divisors in blow-us are examles of cycles that are rigid, in the sense that their reresentative is unique, and hence unwigglable. Still, we can define an algebraic roduct that reduces to the geometric one when transversality can be achieved (see [Ful98]). Examle: the Chow Ring of Projective Sace. A (P n ) = C[H] (H n+ ), where H A (P n ) is the class of a hyerlane H... Chern Classes of Bundles For every vector bundle there is a natural section s 0 B E defined by s 0 (b) = (b, 0) {b} C n. It is called the zero section, and it gives an embedding of B into E. A natural question to ask is if there exists another section s B E which is disjoint form the zero section, i.e. s(b) /= s 0 (b) for all b B. The Euler class of this vector bundle (e(e) A n (B)) is defined to be the class of the self-intersection of the zero section: it measures obstructions for the above question to be answered affirmatively. This means that e(e) = 0 if and only if a never vanishing section exists. It easily follows from the Poincaré-Hof theorem that for a manifold M, the following formula holds: e(t M) [M] = χ(m). That is, the degree of the Euler class of the tangent bundle is the Euler characteristic. The Euler class of a vector bundle is the first and most imortant examle of a whole family of secial cohomology classes associated to a bundle, called the Chern classes of E. The k-th Chern class of E, denoted c k (E), lives in A k (B). In the literature you can find a wealth of definitions for

29 9 Chern classes, some more geometric, dealing with obstructions to finding a certain number of linearly indeendent sections of the bundle, some urely algebraic. Such formal definitions, as imortant as they are (because they assure us that we are talking about something that actually exists!), are not articularly illuminating. In concrete terms, what you really need to know is that Chern classes are cohomology classes associated to a vector bundle that satisfy a series of roerties, which we now recall. Let E be a vector bundle of rank n: identity: by definition, c 0 (E) =. normalization: the n-th Chern class of E is the Euler class: vanishing: for all k > n, c k (E) = 0. c n (E) = e(e). ull-back: Chern classes commute with ull-backs: f c k (E) = c k (f E). tensor roducts: if L and L are line bundles, c (L L ) = c (L ) + c (L ). Whitney formula: for every extension of bundles 0 E E E 0, the k-th Chern class of E can be comuted in terms of the Chern classes of E and E, by the following formula: c k (E) = c i (E )c j (E ). i+j=k Using the above roerties it is immediate to see:. all the Chern classes of a trivial bundle vanish (excet the 0-th);. for a line bundle L, c (L ) = c (L). To show how to use these roerties to work with Chern classes, we calculate the first Chern class of the tautological line bundle over P. The tautological line bundle is S π P,

30 0 where S = {(, l) C P l}. It is called tautological because the fiber over a oint in P is the line that oint reresents. Our tautological family fits into the short exact sequence of vector bundles over P 0 S C P Q 0 P where Q is the bundle whose fibre over a line l P is the quotient vector sace C /l. Notice that Q is also a line bundle. From the above sequence, we have that 0 = c (C P ) = c (S) + c (Q). (.) Since P is toologically a shere, which has Euler characteristic, then = c (T P ) = c (S ) + c (Q) = c (S) + c (Q). (.) The second equality in. holds because T P is the line bundle Hom(S, Q) = S Q. It now follows from (.) and (.) that c (S) =... The Chow ring of M 0,n In [Kee9], Keel constructs M 0,n by a sequence of blow-us along smooth, disjoint, codimension subvarieties starting from M 0,n P. This allows him to give a resentation of the Chow ring. Theorem ([Kee9]). A (M 0,n ) = Z[{D(A)} A [n], A n ]/I R, where {D(A)} denotes the set of boundary divisors and the ideal of relations I R is generated by: symmetry D(A) = D(A c ); disjointness D(A)D(B) = 0 if their set theoretic intersection is emty (in Exercise 8 you were asked for the combinatorial condition characterizing this); WDVV f (0) = f ( ) where f is any forgetful morhism f M 0,n M 0,, and 0, M 0, are two boundary oints. Exercise. Write down exlicitly the combinatorial condition corresonding to the WDVV relations. Intuition. We should interret Keel s theorem as a statement that things just went as well as ossible. Boundary divisors generate the Chow ring, and the relations are only the inevitable ones coming from set theoretic disjointness, lus relations arising from the fact that any two oints in P = M 0, are equivalent.

31 On the one hand, Keel s result is a comlete and conclusive result about the intersection theory of M 0,n, esecially if one is interested mainly in its algebraic structure. If one is more directly interested in the geometry of intersections, then such a resentation is still a bit crytic. It is not immediate that intersections of degree greater than the dimension of M 0,n vanish, for examle. Also, it takes some thought to deal with self intersections of divisors. In the next few exercises we exlore these issues. Exercise. Show that for any collection of n distinct divisors in M 0,n, their roduct vanishes. Exercise. Show that all codimension two strata in M 0, are equivalent. However, show that for each such stratum there is a unique exression as roduct of divisors which is true at the level of cycles. Exercise. Comute the self intersection D({, }) A (M 0, ) as follows: relace one coy of D({, }) using a WDVV relation, and reduce the roblem to a sequence of transverse intersections. The last few exercises hoefully are convincing that one can indeed extract any geometric information from Keel s resentation; however that the rocess is somewhat laborious. This motivates us to introduce new Chow classes, which hel us getting a more direct handle on geometric intersections of strata in M 0,n.. Psi classes For i =,..., n, we define the class ψ i A (M 0,n ). We give a coule seudodefinitions, which are good to develo intuition, before we give a formal one. Take. Let L i M 0,n be a line bundle whose fiber over each oint (C,,..., n ) is canonically identified with T i (C). The line bundle L i is called the i-th cotangent (or tautological) line bundle. Then ψ i = c (L i ). (.6) Take. Let (X, B, π, σ,..., σ n ) be a family of rational stable n-ointed curves, and assume X is smooth. Denote f π the ma to M 0,n corresonding to such a family. Then f π (ψ i ) = c (σ i N σi /X). (.7) Intuition. These two seudo-definitions can be made recise, and they both reflect how one exloits the dictionary between geometry of a moduli sace and geometry of the objects arameterized. We now rovide an bona fide definition of si classes.

32 Take. Define the relative dualizing sheaf ω π U 0,n to be ω π = Ω (U 0,n )/π Ω (M 0,n ). (.8) If we restrict ω π to a fiber of π corresoding to a curve C, we obtain the dualizing sheaf ω. This is a sheaf on C whose sections are meromorhic differential forms on the normalization of C such that they only have oles and at most of order one at the shadows of nodes of C, and the residues at the two shadows of a node must match. The following exercise is the local comutation that is at the base of roving this fact. Exercise. Consider the neighborhood of a node, i.e. the family C (C, t) given by the equation: xy = t. Show that the sections of the sheaf Ω C /π Ω C indeed restrict to the central fiber to meromorhic forms with at most one oles at the node and matching residues there. Definition 9. The class ψ i A (M 0,n ) is defined to be: ψ i = c (σ i ω π ) As a warm-u, consider a divisor of the form D({i, j}) M 0,n, and let us describe the class of its self-intersection. Recall that we can think of M 0,n together with the morhism forgetting the j-th oint as a universal family for M 0,n. Then the divisor D({i, j}) is the image of the section σ i. Using the second seudo-definition, we obtain: D({i, j}) = σ i (e (σ i (N σi /M 0,n ))) = σ i (ψ i ). Exercise 6. Show that on M 0, all si classes vanish (trivially) and on M 0,, for any i we have ψ i = [t.]... Smoothing nodes and the normal bundle to a boundary divisor Consider the gluing morhism: gl I M 0,I M 0,I c M 0,n whose image is the divisor D(I). We describe (ull-back of) the normal bundle N D(I)/M 0,n in terms of tautological bundles on the factors. Intuition. The game is to relate the geometry of the moduli sace with the geometry of the objects arameterized. To describe the normal direction to the divisor D(I) we consider an infinitesimal family of curves, where the central fiber belongs to D(I) and the generic fiber doesn t - which means that the generic fiber is a smooth P. We note that moving out of the divisor corresonds to smoothing the node of the central fiber.

33 Let C t denote a one arameter family of rational stable n-ointed curves, such that C 0 D(I) and C t M 0,n for t /= 0. We can interret t as a coordinate in a direction normal to D(I) at C 0. Since we are interested in infinitesimal (first order) information, we can look at the local analytic equation for C t around the node of the central fiber, and notice that it has the form xy = t. We observe that x and y can be identified with sections of the tangent saces of the two axes in C. These are the tangent saces at the shadows of the node in the normalization of C 0. Finally we observe that once we give this interretation, the relation rovided by the local equation xy = t gives a relation among sections of line bundles on D(I). This is an imressionistic argument for the following statement. Lemma. Consider the gluing morhism gl I M 0,I M 0,I c M 0,n. Then : gl I (N D(I)/M 0,n ) L L. (.9) Lemma allows us to describe the self-intersection of a boundary divisor in terms of si classes. Lemma. Consider the gluing morhism gl I M 0,I M 0,I c M 0,n and let π and π denote the two rojections from M 0,I M 0,I c onto the factors. Then: D(I) = gl I ( π (ψ ) π (ψ )). (.0) Exercise 7. Recomute the self intersection D({, }) A (M 0, ) using Lemma. Exercise 8. Describe a combinatorial algorithm that determines the intersection of any two boundary strata in M 0,n in terms of boundary strata and si classes... Proerties of si classes In Section.. we showed that we can understand non-transversal intersection of boundary strata in terms of si classes. This motivates us to seek for a better understanding of these classes. Because of Keel s resentation, we know that si classes must be equivalent to a linear combination of boundary divisors, but is there a nice descrition for such a linear combination? Also, what if we want to deal with a roduct of si classes? In an extreme case, suose we are given a monomial in si classes of degree n, which is just a multile of the class of a oint - is there a quick way to determine this multile? In this section we answer all these questions in the ositive. The first roerty of si classes we exlore is how they restrict to boundary divisors (or strata in general). Informally, a boundary stratum is a roduct of moduli saces; the restriction of ψ i to the stratum equals the class ψ i ulled back from the factor containing the i th mark. More formally:

34 Lemma. Consider the gluing morhism gl I M 0,I M 0,I c M 0,n. Assume that i I and denote by π M 0,I M 0,I c M 0,I the first rojection. Then: gl (ψ i ) = π (ψ i ). Intuition. This lemma is obvious if one accets our first seudo-definition of si classes, as then the class ψ i only knows what haens in an infinitesimal neighborhood of the oint i. We leave it to the interested reader to convert this idea into an actual roof. Exercise 9. Prove that ψ i D({i, j}) = 0. The next result is a comarison between a si class on M 0,n+ and the ull-back of the corresonding si class on M 0,n. This result is essential in giving an inductive descrition of si classes. Lemma. Consider the forgetful morhism π n+ M 0,n+ M 0,n. Then, for every i =,..., n, ψ i = π n+(ψ i ) + D({i, n + }). In the sirit of this section, instead of a rigorous roof we rovide a coule of quasi-roofs that have the advantage of illustrating what is going on. Using the first definition of ψ i we observe that the fibers of L i and π n+ (L i) are canonically identified excet along the divisor D({i, n + }). Hence it must be that L i π n+(l i ) O(kD({i, n + })), for some integer k. Taking the first Chern class: ψ i = π n+(ψ i ) + kd({i, n + }). (.) We determine that k = in the following exercise. Exercise 0. Determine the value of k in two different ways:. Multily relation (.) by D({i, n + });. Multily relation (.) by the boundary stratum consisting of a chain of rojective lines, with the oints, i and n + on the first twig of the chain, and exactly three secial oints on each other twig.

35 Using the second definition of ψ i, we consider a one arameter family of n ointed curves (X, B, π, σ,..., σ n ). We now consider an arbitrary section σ n+, not necessarily disjoint from the other sections (but let us assume that the intersections σ n+ σ i for all i are transversal). The family of rational stable (n + ) ointed curves ( ˆX, B, ˆπ, ˆσ,..., ˆσ n, ˆσ n+ ) is obtained by blowing u the intersection oints of σ n+ with the other sections, and taking the roer transforms of the sections. Denote fˆπ B M 0,n+ the corresonding ma to the moduli sace. Identifying B with σ i (B) we have that Identifying B with ˆσ i (B), Finally, we use the fact that σ i (B) = f ˆπ (π n+(ψ i )). ˆσ i (B) = f ˆπ (ψ i). σ = ˆσ + E i,n+, where E i,n+ is the excetional divisor correonding to coies of P maing to each of the intersection oints of σ i with σ n+. It follows that for any one arameter family fˆπ, f ˆπ (ψ i) = f ˆπ (π n+(ψ i ) + D({i, n + })). (.) Lemma allows a combinatorial descrition of cycles reresenting the si classes. Exercise. Recover that on M 0,, ψ i = [t.] by using Lemma. Describe cycles reresenting ψ i in M 0,. Once you have warmed u, you can then rove the following general statement. Lemma 6. For any choice of i, j, k distinct, we have the following equation in A (M 0,n ): ψ i = i I, j,k/ I D(I). Show that any two such exressions for ψ i are WDVV equivalent. Exercise. Prove that for any k. ψ k i = π n+(ψ i ) k (π n+(ψ i ) + D({i, n + })). Now we investigate how a monomial in si classes ushes forward with resect to a forgetful morhism.

36 6 Lemma 7 (String Equation). Consider the forgetful morhism π n+ M 0,n+ M 0,n. Then n π n+ ( i= ψ k i i ) = j k j /=0 ψ k j j ψ k i i. i/=j Exercise. Prove Lemma 7, by rewriting each of the ψ k i i as in Exercise, and observing which terms in the exansion of such roduce do not vanish immediately, or after the ush-forward. Lemma 7 allows us to exlicity evaluate all to degree monomials in si classes. Exercise. Let k i = n. Then n+ ψ k n i i = ( ), M 0,n i= k,..., k n+ where the integral sign denotes ush-forward to the class of a oint. Exercise. Consider the forgetful morhism π n+ M 0,n+ M 0,n. What is π n+ (ψ n+ )?.6 Further exercises Exercise 6. Consider the two models of M 0, we have constructed: and Bl (0,0),(,),(, ) (P P ) Bl (0 0 ),(0 0),(0 0 ),( ) (P ). In each of the two cases identify all the boundary strata. Exercise 7. The boundary comlex is the cone comlex naturally associated to the oset structure of the boundary strata in M 0,n. For each codimension i stratum you have a coy of (R 0 ) i, and the oset structure naturally indicates how to identify cones with faces of other cones. Make this statement recise, and describe the boundary comlex of M 0,. If you are familiar with Peterson grah, recognize that it is the cone over the Peterson grah. Exercise 8 (Constructions of M 0,n ). This exercise is not so much an exercise, as it is a quick and dirty sketch of some constructions of M 0,n, in hoe that it may hel you read the relative literature if you decide to dig deeer into it. Try to follow these constructions in the cases n =, and get a feel of why they work in general.

37 7 Knudsen [Knu8] Knudsen s construction is inductive, and it relies essentially on the identification of U 0,n with M 0,n+ given by the contraction and stabilization morhisms. Suose we have realized M 0,n as a fine moduli sace, which means in articular that we have constructed the universal family U 0,n. We then take M 0,n+ to be equal to U 0,n, and we must construct a universal family over it. First consider the fiber roduct of M 0,n+ with itself over M 0,n. This gives a family of n ointed curves X. We can add one more section to this family by considering the diagonal section =. Then we stabilize this family, to obtain a family of rational stable (n + )-ointed curves over M 0,n+. This is the universal family. U 0,n+ Bl δ=σi ˆσ i ˆδ X σ i M 0,n+ δ M 0,n+ M 0,n σ i Keel [Kee9] Keel s construction is in a way similar to Knudsen, in that it starts from a fiber roduct of M 0,n+ over M 0,n to get a family of n-ointed curves over M 0,n+. His fiber roduct is with M 0, so that now the n sections intersect wildly. Next he adds the n + -th section just like Knudsen. Finally he resolves in stes the intersections of the sections, by iteratively blowing u loci where the largest number of sections intersect. By doing so he is insuring that he is always blowing u along smooth codimension two centers. U 0,n+ = B n... B σ i M 0,n+ δ M 0,+ M 0,n Karanov [Ka9b] Karanov s construction identifies M 0,n with the family R M 0,n P n of rational normal curves in P n through n general oints. Then he considers the closure of such family R M 0,n P n to be the universal family over M 0,n and, hence, M 0,n+.

38 8 This construction constructs M 0,n+ as a sequence of blowus along smooth centers: first one blows u the n oints in general osition, then the roer transforms of the lines joining each air of oints, then the roer transforms of the lanes containing any three such oints, etc... Exercise 9. Consider the forgetful morhism π n+ M 0,n+ M 0,n. We define: π n+ (ψ i+ n+) = κ i. Kaa classes are imortant in the theory of moduli saces of higher genus curves, as they give us interesting classes that exists on the moduli sace of unointed curves, and that are not entirely suorted on the boundary. In the genus zero theory everything is boundary, and you need at least three oints. For this reason I am not aware of these classes laying a rominent role in this theory. However a erfectly sensible question, whose answer I couldn t find with a quick literature search, is whether there is a good combinatorial reresentation of kaa classes in terms of boundary strata (along the lines of the descrition of si classes in Lemma??).