Enumeration of One-Nodal Rational Curves in Projective Spaces

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1 Enumeration of One-Nodal Rational Curves in Projetive Spaes Aleksey Zinger June 6, 2004 Abstrat We give a formula omputing the number of one-nodal rational urves that pass through an appropriate olletion of onstraints in a omplex projetive spae. The formula involves intersetions of tautologial lasses on moduli spaes of stable rational maps. We ombine the methods and results from three different papers. Contents 1 Introdution 1 2 Bakground Topology Notation A Strutural Desription Computations Summary and Motivation A Tree of Chern Classes Combinatoris Comparison of n 1 d µ and n 1,dµ Summary Dimension Counts A Property of Limits in M 1,N P n,d Introdution Enumerative algebrai geometry is a field of mathematis that dates bak to the nineteenth entury. However, many of its most fundamental problems remained unsolved until the early 1990s. For Partially supported by NSF grant DMS

2 example, let d be a positive integer and µ = µ 1,...,µ N an N-tuple of linear subspaes of P n of odimension at least two suh that l=n odim C µ odim C µ l N = dn n 3. l=1 If the onstraints µ are in general position, denote by n d µ the number of rational degree-d urves that pass through µ 1,...,µ N. This number is finite and depends only on the homology lasses of the onstraints. If d = 1, it an be omputed using Shubert alulus; see [GH]. All but verylow-degree numbers n d µ remained unknown until [KM] and [RT] derived a reursive formula for these numbers. In this paper, we prove Theorem 1.1 Suppose n 3, d 1, and µ = µ 1,...,µ N is an N-tuple of proper subvarieties of P n in general position suh that l=n odim C µ odim C µ l N = dn l=1 Then the number of degree-d rational urves that have a simple node and pass through the onstraints µ is given by n 1 d µ = 2 1 RT1,d µ 1 ;µ 2,...,µ N CR 1 µ, where 2k n+1 n+1 2k n+1 a CR 1 µ = 1 k 1 k 1! l η [ Vk n+1 2k l, µ ]. l k=1 l=0 The sympleti invariant RT 1,d ; and the top intersetions a l η n+1 2k l, [ Vk µ ] are omputable via algorithms desribed elsewhere. n d µ 5,5 5,1,4 5,1,0,4 2,1,1,7 2,1,1,1,6 n 1 d µ 1,800 1,800 1,800 20,340 20,340 Table 1: The Number n 1 d µ of One-Nodal Degree-d Rational Curves in Pn For the purposes of this table, we assume that the onstraints µ 1,...,µ N are linear subspaes of P n of odimension at least two. We desribe suh a tuple µ of onstraints by listing the number of linear subspaes of odimension 2,...,n among µ 1,...,µ N. For example, the triple 5,1,4 in the third olumn indiates that the tuple µ onsists of 5 two-planes, 1 line, and 4 points in general position in P 4. In the statement of Theorem 1.1, RT 1,d ; denotes the genus-one degree-d sympleti invariant of P n defined in [RT]. This invariant an be expressed in terms of the numbers n d ; see [RT]. In 2

3 partiular, it is omputable. Brief remarks onerning the meaning of RT 1,d ; an be found at the beginning of Setion 3. The ompat oriented topologial manifold V k µ onsists of unordered k-tuples of stable rational maps of total degree d. Eah map omes with a speial marked point i,. All these marked points are mapped to the same point in P n. In partiular, there is a well-defined evaluation map ev: V k µ P n, whih sends eah tuple of stable maps to the value at one of the speial marked points. We also require that the union of the images of the maps in eah tuple interset eah of the onstraints µ 1,...,µ N. In fat, the elements in the tuple arry a total of N marked points, y 1,...,y N, in addition to the k speial marked points. These marked points are mapped to the onstraints µ 1,...,µ N, respetively. Roughly speaking, eah element of V k µ orresponds to a degree-d rational urve in P n, whih has at least k irreduible omponents, and k of the omponents meet at the same point in P n. The preise definition of the spaes V k µ an be found in Subsetion 2.2. The ohomology lasses a and η l are tautologial lasses in V k µ. In fat, a = ev 1 OP n1. Let V k µ be the oriented topologial manifold defined as V k µ, exept without speifying the marked points y 1,...,y N mapped to the onstraints µ 1,...,µ N. Then, there is well-defined forgetful map, π: V k µ V k µ, whih drops the marked points y 1,...,y N and ontrats the unstable omponents. The ohomology lass η l H 2l V k µ is the sum of all degree-l monomials in the elements of the set π ψ 1,,...,π ψ k, } H 2 Vk µ. As ommon in algebrai geometry, ψ i, denotes the first hern lass of the universal otangent line bundle for the marked point i,. In Subsetion 2.2, we give a definition of η l that does not involve the projetion map π. An algorithm for omputing the intersetion numbers involved in the statement of Theorem 1.1 is given in Subsetion 5.7 of [Z2]. It is losely related to the algorithm of [P2] for omputing intersetions of tautologial lasses in moduli spaes of stable rational maps into P n. If n=2, we denote by n 1 d µ the number of rational degree-d urves passing through the onstraints ounted with a hoie of the node on eah urve. The formula of Theorem 1.1 gives n 1 d 1 d µ = n d µ This identity is lear, sine the arithmeti genus of every degree-d urve in P 2 is d 1 2. Equation 1.2 is used in [P1] to ount genus-one plane urves with omplex struture fixed. More preisely, if µ is a tuple of onstraints in P n satisfying ondition 1.1, let n 1,d µ denote the number of genus-one degree-d urves that pass through the onstraints µ and have a fixed generi 3

4 omplex struture on the normalization, i.e. its j-invariant is different from 0 and The key step in [P1] is to show that n 1,d µ = n 1 d µ, 1.3 if µ is a tuple of 3d 1 points in P 2. One of the main ingredients in proving Theorem 1.1 is Proposition 4.1, whih states that 1.3 is valid for any tuple µ that satisfies ondition 1.1. Note that the numbers listed in Table 1 are onsistent with 1.3 and fats of lassial algebrai geometry. In partiular, the image of every degree-4 map from a genus-one urve to P n lies in a P 3 and the image of every degree-6 map lies in a P 5 ; see [ACGH, p116]. Thus, the first three numbers in the table should be the same, and the last two numbers should be the same. The proof of Proposition 4.1 extends the degeneration argument of [P1] and builds up on modifiations desribed in [Z1]. We work with the moduli spae M 1,N P n,d of stable degree-d maps from genus-one N-pointed urves into P n and study what happens in the limit to the maps that pass through the onstraints µ as the j-invariant of the domain tends infinity, i.e. the domain degenerates to a rational urve with two points identified. Proposition 4.1 is not useful for determining the numbers n 1,d µ in P n if n 3, sine the right-hand side of 1.3 is unknown. Computation of n 1,d µ for all projetive spaes is the subjet of [I], where an entirely different approah is taken. The main step in omputing these numbers is showing that 2n 1,d µ = RT 1,d µ 1 ;µ 2,...,µ N CR 1 µ, where CR 1 µ is the number of zeros of an expliit affine map between vetor bundles over V 1 µ; see Proposition 3.2. The remaining step is to express this number of zeros topologially. In general, if the linear part of an affine map ψ does not vanish, it is easy to determine the signed ardinality of ψ 1 0; see Lemma 2.5. The approah of [I] is to replae the linear part α of the affine map under onsideration by a nonvanishing linear map over a spae obtained from V 1 µ by a sequene of blowups and then to express the resulting intersetion number in terms of intersetion numbers on the spaes V k µ. The main problem with this approah is that the new linear map is not desribed in [I] and it is not lear how to onstrut it in general. In addition, the normal bundles of ertain spaes needed for the seond part of this approah are given inorretly; see Lemma 2.8 or equation 2.27 in [I] for example. Both of these statements an be orreted without affeting the omputability of the intersetion numbers, but presumably with a hange in the final result. If n=2, no blowup is needed. If n=3,4, the zero set of α is a omplex manifold and the derivative of α in the normal diretion along α 1 0 is nondegenerate. In suh ases, only one blowup is needed and a linear map with the required properties an be onstruted fairly easily. Furthermore, Lemma 2.8 of [I] requires no orretion in the n=2,3,4 ases, while equation 2.27 is never used. If n=2,3, CR 1 µ and n 1,d µ are then expressed in terms of the numbers n d µ, with d d and µ related to µ. Several numbers n 1,d µ for P 4 are given in [I] as well. However, no topologial formula, like that of Theorem 1.1, is given for CR 1 µ or n 1,d µ for P n with n 4 and no number n 1,d µ is given for P n with n 5. We obtain the expression of Theorem 1.1 for the number CR 1 µ in Setion 3; see Proposition 3.1. Our approah involves no blowups and requires relatively little understanding of the global struture of the spaes V k µ. Instead we desribe CR 1 µ as the euler lass of a bundle minus the sum of ontributions to the euler lass from smooth, but usually nonompat, strata of the zero set of the linear part α 1,0 of the affine map. Computation of these ontributions in good ases involves 4

5 ounting the zeros of affine maps again, but with the rank of the target bundle redued by one; see Subsetion 2.1. Of ourse, if we are to have any hope of omputing these ontributions, we need to understand the behavior of α 1,0 near the smooth strata of its zero set. Proposition 2.7 desribes the behavior of α 1,0 and of related linear maps near the boundary strata of V k µ. Theorem 1.1 follows immediately from Propositions 3.1 and 4.1. Their proofs are mutually independent. Setion 4 uses some of the notation defined in Subsetion 2.2. The topologial tools of Subsetion 2.1, the desriptive notation of Subsetion 2.2, and the struture theorem of Subsetion 2.3 are integral to the omputations of Setion 3. In brief, we enumerate one-nodal rational urves from genus-one fixed-omplex-struture invariants. Can a similar approah be used with higher-genera enumerative invariants? Let µ be an N-tuple of proper subvarieties of P n in general position suh that odim C µ = dn + 1 n. Denote by n 2,d µ the number of genus-two degree-d urves that pass through the onstraints µ and have a fixed generi omplex struture on the normalization. Let n 3 d µ, τ dµ, and T d µ denote the number of rational two-omponent urves onneted at three nodes, of rational urves with a triple point, and of rational urves with a tanode, respetively. If n = 2, we take n 3 d µ to be the number of two-omponent rational urves with a hoie of three nodes ommon to both omponents. In all ases, the urves have degree-d and pass through the onstraints µ. Completing the degeneration argument of [KQR], it is shown in [Z1] that n 2,d µ = 6 n 3 d µ + τ dµ + T d µ, 1.4 if µ is a tuple of 3d 2 points in P 2. The arguments of [KQR] and [Z1] should extend to show that equation 1.4 is valid for arbitrary onstraints µ in all projetive spaes. On the other hand, n 2,d µ for P 3 is omputed in [Z2] and the method extends at least to P 4. Thus, in those two ases, we should be able to express the sum of the numbers n 3 d µ, τ dµ, and T d µ in terms of intersetion numbers of the spaes V k µ. The relation 1.4 is obtained by onsidering a degeneration to a speifi singular genus-two urve. Perhaps, different relations an be obtained by onsidering degeneration to other singular genus-two urves. With enough different relations, we would be able to ompute the numbers n 3 d µ, τ dµ, and T d µ at least for P 3 and P 4. Sine the initial submission of this paper, a formula for the numbers n 1 d µ in P3, i.e. the lowestdimensional ase of Theorem 1.1, has also appeared in [R]. The approah of [R] is ompletely unrelated to the one presented here; it uses more lassial tools of algebrai geometry, instead of the moduli spae of stable maps. The author thanks T. Mrowka for many useful disussions and E. Ionel for omments on the original version of this paper. 5

6 2 Bakground 2.1 Topology We begin by desribing the topologial tools used in the next setion. In partiular, we review the notion of ontribution to the euler lass of a vetor bundle from a not neessarily losed subset of the zero set of a setion. We also reall how one an enumerate the zeros of an affine map between vetor bundles. These onepts are losely intertwined. Details an be found in Setion 3 of [Z2], where these onepts are presented in a greater generality. Throughout this paper, all vetor bundles are assumed to be omplex and normed. If F M is a smooth vetor bundle, losed subset Y of F is small if it ontains no fiber of F and is preserved under salar multipliation. If Z is a ompat oriented zero-dimensional manifold, we denote the signed ardinality of Z by ± Z. If k is an integer, we write [k] for the set of positive integers not exeeding k. Definition 2.1 Suppose F, O M are smooth vetor bundles, Ω is an open subset of F, and φ: Ω O is a smooth bundle map. 1 Bundle map α: F O is a dominant term of φ if there exists ε C 0 F; R suh that φυ αυ ευ αυ υ Ω and lim ευ = 0. υ 0 2 The dominant term α: F O of φ is the resolvent of φ if α: F O is linear map whih is injetive on every fiber of F. 3 The bundle map φ: Ω O is hollow if there exist a vetor bundle F M of rank less than the rank of F, a smooth bundle map ρ: F F, and a linear map α: F O, whih is injetive on every fiber, suh that α ρ is a dominant term of φ. If F M is a vetor bundle, we denote by γ F PF the tautologial line bundle and by π PF :PF M the bundle projetion map. If α is a setion of the bundle HomF, O, let α be the setion of Homγ F,πPF O indued by α. The base spaes we work with in the next two setions are losely related to spaes of rational maps into P n of total degree d that pass through the N onstraints µ 1,...,µ N. From the algebrai geometry point of view, spaes of rational maps are algebrai staks, but with a fairly obsure loal struture. We view these spaes as mostly smooth, or ms-, manifolds: ompat oriented topologial manifolds stratified by smooth manifolds, suh that the boundary strata have real odimension at least two. Subsetion 2.3 gives expliit desriptions of neighborhoods of boundary strata and of the behavior of ertain bundle setions near suh strata. We all the main stratum M of ms-manifold M the smooth base of M. Definition 3.7 in [Z2] also introdues the natural notions of ms-maps between ms-manifolds, ms-bundles over ms-manifolds, and ms-setions of ms-bundles. Definition 2.2 Let M=Mn n 2 i=0 M i=m n 2 i=0 M i be an ms-manifold of dimension n. 1 If Z M i is a smooth oriented submanifold, a normal-bundle model for Z is a tuple F,Y,ϑ, where 1a F Z is a smooth vetor bundle and Y is a small subset of F; 1b for some δ C Z; R +, ϑ: F δ Y Z M is a ontinuous map suh that 6

7 1b-i ϑ: F δ Y Z M is a homeomorphism onto an open neighborhood of Z in M Z; 1b-ii ϑ Z is the identity map, and ϑ: F δ Y Z M is an orientation-preserving diffeomorphism on an open subset of M. 2 A losure of normal-bundle model F,Y,ϑ for Z is a tuple Z,F,π, where 2a Z is an ms-manifold with smooth base Z; 2b π: Z M is an ms-map suh that π Z is the identity map; 2 F Z is an ms-bundle suh that F Z =F. We use a normal-bundle model for Z to desribe the behavior of bundle setions over M near Z. Eah setion we enounter in this paper exhibits one of the two kinds of behavior desribed by Definition 2.3. Definition 2.3 Suppose M is an ms-manifold, V M is an ms-bundle, s Γ M;V, and Z s Z is s-hollow if there exist a normal-bundle model F,Y,ϑ for Z and a bundle isomorphism ϑ V : ϑ V π F V, overing the identity on F δ Y Z, suh that 1a ϑ V Fδ Y Z is smooth and ϑ V Z is the identity; 1b the map φ ϑ V ϑ s: F δ Y Z V is hollow. 2 Z is s-regular if there exist a normal-bundle model F,Y,ϑ for Z with losure Z,F,π, setion α Γ Z,HomF,π V, and a bundle isomorphism ϑ V : ϑ V π F V overing the identity on F δ Y Z, suh that 2a ϑ V Fδ Y Z is smooth and ϑ V Z is the identity; 2b α Z is nondegenerate and is the resolvent for φ ϑ V ϑ s: F δ Y Z V ; 2 the spae PF admits a deomposition into subspaes Z i } suh that eah spaes Z i is either α-hollow or satisfies 2a and 2b with s replaed by α. If M is a smooth manifold and Z is a smooth ompat submanifold of M suh that s vanishes along Z, but the derivative of s in the normal diretion along Z is nondegenerate, Z is s-regular. The full-rank linear map α is the derivative of the setion s in the normal diretion along Z. However, if the derivative of s in the normal diretion does not have full rank, Z may not be s-hollow. For example, if s is the setion of the trivial line bundle over C given by sz=z 2, the submanifold 0} is not s-hollow. In fat, 0} is s-regular in the sense of Setion 3 in [Z2]. On the other hand, if s is the setion of the trivial rank-two bundle over C C given by C C C C, sz,w=zw,zw 2, 0} is s-hollow, while the submanifold 0} C is s-regular. In ontrast, the submanifold 0} C is not s-regular. We all s Γ M;V a regular setion if M an be omposed into s-hollow and s-regular subspaes. We all α Γ Z;HomF, O a regular linear map if α satisfies the requirements of 2 of Definitions 2.3. If α Γ M;HomE, O is a linear map and rke+ 1 2 dim M=rk O, the zero set of the affine map ψ α, ν : E O, ψ α, ν υ = ν υ + αυ, 7

8 is a zero-dimensional oriented submanifold of E M, if ν Γ M; O is a generi setion; see Lemma 3.10 in [Z2]. If α is a regular linear map, ψα, ν0 1 is a finite set for a generi hoie of ν, and the number Nα ± ψ 1 α, ν0 is independent of suh a hoie of ν. We are now ready to state the first part of the omputational method of this paper, Proposition 2.4. The seond part is Lemma 2.5. Proposition 2.4 Let V M be an ms-bundle of rank n over an ms-manifold of dimension 2n. Suppose U is an open subset of M and s Γ M;V is suh that s U is transversal to the zero set. 1 If s 1 0 U is a finite set, ± s 1 0 U = ev,[ M] C M U s. i=k 2 If M U = Z i, where eah Z i is s-regular or s-hollow, then s 1 0 U is finite, and i=1 ± s 1 0 U = ev,[ M] C M U s = ev,[ M] i=k C Zi s. If Z i is s-hollow, C Zi s=0. If Z i is s-regular and α i : F i π i V is the orresponding linear map, C Zi s = Nα i. Finally, if α i Γ Z i ;HomF i,π i V has full-rank rank over all of Zi, C Zi s = πi V F i 1,[ Z i ]. This proposition is a speial ase of Corollary 3.13 in [Z2]. Proposition 2.4 redues the problem of omputing C Zi s for an s-regular manifold Z i to ounting the zeros of an affine map between two vetor bundles. The general setting for the latter problem is the following. Suppose E, O M are ms-bundles, suh that rke+ 1 2 dim M = rk O, and α : E O is a regular linear map. Let ν Γ M; O be suh that the map ψ α, ν ν+α: E O is transversal to the zero set in O on E M, and all its zeros are ontained in E M. Then Nα ± ψ 1 α, ν0 depends only on α. If the rank of E is zero, then learly Nα = ± ψ 1 α, ν0 = eo,[ M]. If the rank of E is positive and ν is generi, the setion ν does not vanish and thus determines a trivial line subbundle C ν of O. Let O = O/C ν and denote by α the omposition of α with the quotient projetion map. If E is a line bundle and α is a linear map, i=1 Nα = ± ψ 1 α, ν0 = ee O,[ M] C α 1 0α. By Proposition 2.4, omputation of C α 1 0α again involves ounting the zeros of affine maps, but with the rank of the new target bundle, i.e. E O, one less than the rank of the original one, i.e. O. On the other hand, if the rank of E is bigger than one, Nα = N α; see Subsetion 3.3 in [Z2]. Thus, at least in reasonably good ases, the number Nα an be determined in finitely many steps. The next lemma summarizes the results of Subsetion 3.3 in [Z2]. Let λ E = 1 γ E H2 PE. 8

9 h ι h ˆ0 Figure 1: A Rooted Tree Lemma 2.5 Suppose M is an ms-manifold and E, O M are ms-bundles suh that rk E dim M = rk O. If α Γ M;HomE, O and ν Γ M; O are suh that α is regular, ν has no zeros, the map ψ α, ν ν+α: E O is transversal to the zero set on E M, and all its zeros are ontained in E M, then ψ 1 α, ν0 is a finite set, ± ψ 1 α, ν0 depends only on α, and Furthermore, if n=rk E, Nα ± ψ 1 α, ν 0 = OE 1,[ M] C α 1 0 α. k=n λ n E + k Eλ n k E = 0 H 2n PE and k=1 µλ n 1 E,[PE] = µ,[ M] µ H 2m 2n M Notation In this subsetion, we desribe the most important notation used in this paper. Some of the notation is only skethed; see Setion 2 in [Z3] for more details. Let q N : C S 2 R 3 be the stereographi projetion mapping the origin in C to the north pole. We identify C with S 2 } via the map q N, where Let e =1,0,0 T S 2. = 0,0, 1 S 2 R 3. Definition 2.6 A finite partially ordered set I is a linearly ordered set if for all i 1,i 2,h I suh that i 1,i 2 <h, either i 1 i 2 or i 2 i 1. A linearly ordered set I is a rooted tree if I has a unique minimal element, i.e. there exists ˆ0 I suh that ˆ0 i for all i I. In Figure 1, the dots denote the elements of a rooted tree I and the arrows desribe the partial ordering. If I is a linearly ordered set, let Î be the subset of the non-minimal elements of I. For every h Î, denote by ι h I the largest element of I whih is smaller than h; see Figure 1. Suppose I = I k is the splitting of I into rooted trees suh that k is the minimal element of I k. If ˆ1 I, k K 9

10 ˆ1 k 1 k 2 k 1 k 2 Figure 2: Linearly Ordered Sets I and I+ k1ˆ1 we define the linearly ordered set I+ kˆ1 to be the set I+ˆ1 with all partial-order relations of I along with the relations k<ˆ1 and ˆ1<h if h Îk; see Figure 2. If S is a possibly singular omplex urve and M is a finite set, a P n -valued bubble map with M-marked points is a tuple b = S,M,I;x,j,y,u, where I is a linearly ordered set, and x: Î S S2, j : M I, y: M S S 2, and u: I C S; P n C S 2 ; P n are maps suh that S 2 }, if ι h x h Î; S, if ι h Î; y l S 2 }, if j l Î; S, if j l Î; u i C S 2 ; P n, if i Î; C S; P n, if i Î; and u h =u ιh x h for all h Î. We assoiate suh a tuple with Riemann surfae i} S Σ b = Σ b,i /, 2, if where Σ b,i = i Î; i} S, if i Î, and h, ι h,x h h Î, i I with marked points j l,y l Σ b,jl, and ontinuous map u b : Σ b P n, given by u b Σ b,i = u i for all i I. We require that all the singular points of Σ b and all the marked points be distint. Furthermore, if S = S 2, all these points are to be different from eah of the speial marked points i, Σ b,i, where i is a minimal element of I, i.e. one of the elements of the set I Î. In addition, if Σ b,i =S 2 and u i [S 2 ]=0 H 2 P n ; Z, then Σ b,i must ontain at least two singular and/or marked points of Σ b other than i,. If S S 2, but S is unstable, u i must satisfy a similar stability ondition whenever Σ b,i =S. In partiular, if S is a torus or a irle of spheres and the restrition of u i to a omponent S h of S is homologially zero, S h ontains at least one marked point of Σ b. Two bubble maps b and b are equivalent if there exists a homeomorphism φ: Σ b Σ b suh that u b =u b φ, φj l,y l =j l,y l for all l M, φ Σ b,i is holomorphi for all i I, and φσ b,i Σ b,i for some i I Î if i I Î. The general struture of bubble maps is desribed by tuples T =S,M,I;j,d, with d i Z speifying the degree of the map u b on Σ b,i. We all suh tuples bubble types. Bubble type T is simple if I is a rooted tree; T is basi if Î = and d i 0 for all i I; T is semiprimitive if ι h Î, d ι h =0, and d h 0 10

11 for all h Î. The above equivalene relation on the set of bubble maps indues an equivalene relation on the set of bubble types. For eah h, i I, let D i T = h Î : i<h}, Di T = D i T i}, H i T = h Î : ι h=i}, M i T = l M : j l =i}, 0, if i I s.t. h D i T, d i =0; χ T h = 1, if i I s.t. h D i T, d i =0, but d h 0; χt = h I : χ T h=1 }. 2, otherwise; Denote by H T the spae of all holomorphi bubble maps with struture T. The automorphism group of every bubble type T we enounter in the next two setions is trivial. Thus, every bubble type disussed below is presumed to be automorphism-free. If S is a irle of spheres, we denote by M T the set of equivalene lasses of bubble maps in H T. For eah bubble type T = S 2,M,I;j,d, let U T = [b]: b= S 2,M,I;x,j,y,u H T, u i1 = u i2 i 1,i 2 I Î}. Then there exists B T H T suh that U T is the quotient of a subset B T of H T by a G T S 1 I - ation. Denote by U 0 T the quotient of B T by G T S 1 Î G T. Then U T is the quotient of U 0 T by the residual G T S1 I Î G T ation. Corresponding to these quotients, we obtain line orbi-bundles L i T U T : i I }. Let FT = h Î F h T U T, where F h T = L h T L ι h T. Denote by F T the open subset of FT onsisting of vetors with all omponents nonzero. The Gromov-onvergene topology on the spae of equivalene lasses of bubble maps indues a partial ordering on the set of bubble types and their equivalene lasses suh that the spaes Ū 0 T = T T U 0 T and Ū T = are ompat and Hausdorff. The G T -ation on U0 T extends to an ation on Ū0 T, and thus the line orbi-bundles L i T U T with i I Î extend over Ū T. These bundles an be identified with the universal tangent line bundles for appropriate setions of the universal bundle over ŪT. The evaluation maps ev l : H T P n, ev l S,M,I;x,j,y,u = ujl y l, T T U T desend to all the quotients and indue ontinuous maps on ŪT and Ū0 T. If µ=µ M is an M-tuple of subvarieties of P n, let M T µ = b M T : ev l b µ l l M } and define spaes U T µ, ŪT µ, et. in a similar way. If S = S 2, we define another evaluation map, ev: B T P n by ev S 2,M,I;x,j,y,u = uˆ0, 11

12 k k l 1 l 2 Figure 3: The Domains of Elements of U T and U T M0 where ˆ0 is any minimal element of I. This map desends to U 0 T and U T. If µ=µ M is an M-tuple of onstraints, let U T µ = b U T : ev l b µ l l M M, evb µ l l M M } and define U 0 T µ, et. similarly. Suppose T =S 2,M,I;j,d is a bubble type, k I Î, and M 0 is nonempty subset of M k T. Let T /M 0 = S 2,I,M M 0 ;j M M 0,d. Define T M 0 S 2,M,I + k ˆ1;j,d by k, if l M 0 ; j l = ˆ1, if l M k T M 0 ; j l, otherwise; 0, if i=k; d i = d k, if i=ˆ1; d i, otherwise. The tuples T /M 0 and T M 0 are bubble types as long as d k 0 or M 0 M k T. In Figure 3, we show the domain of an element of the spae U T, where I =k} is a single-element set, and the domain of an element of the spae U T M0, where M 0 =l 1,l 2 } is a two-element set. In this and later figures, we denote eah omponent of the domain by a disk and shade the omponents on whih the map into P n is nononstant. We indiate marked points on the ghost omponents, i.e. the omponents on whih the map is onstant, by putting small dots on the boundary of the orresponding disk. The point labeled by k, i.e. the same way as the omponent, is the speial marked point k,. Proposition 2.7 and Lemma 2.8, as well as the deomposition 2.4, show that it is ruial to learly distinguish between ghost and non-ghost omponents. Note that Ū T M0 µ = M ˆ1} M 0 ŪT /M 0 µ, 2.2 where M ˆ1} M 0 denotes the Deligne-Mumford moduli spae of rational urves with ˆ0,ˆ1} M 0 - marked points. If T is a basi bubble type, let 1 L k T 1L k T PD ŪT µ[ūt M0 µ ] H 2 Ū T µ. 2.3 =M 0 M k T This ohomology lass is well-defined; see Subsetion 5.2 in [Z2]. Whenever the bubble type T is lear from ontext, we will write 1 L k and 1L k for 1L k T and 1L kt, respetively. We illustrate definition 2.3 in Figure 4 in the ase I = k} is a single-element set. In this figure, as well in the future ones, we denote spaes of tuples of stable maps by drawing a piture of the 12

13 1 L k k = 1 L k =M k T k Figure 4: An Example of Definition 2.3 k M 0 domain of a typial element of suh a spae. We are now ready to explain the laim of Theorem 1.1. Let n, d, N, and µ be as in the statement of the theorem. If k 1 and m 1, denote by V k,m µ the disjoint union of the spaes ŪT µ taken over equivalene lasses of basi bubble types T =S 2,[N] M 0,I;j,d with M 0 = m, I = k, d i >0, and d i =d. Let V k µ= V k,0 µ. We define the spaes V k,m µ similarly. Let 1 L i: i [k] }, 1 L i : i [k] } H 2 Vk,m µ; Z be given by 1 L i Ū T µ: i [k] } = 1 L i T : i I}, 1 L i Ū T µ: i [k] } = 1 L i T : i I}, where T is as above and [k] = 1,...,k}. We denote by η l, η l H 2l Vk,m µ; Z the sum of all degree-l monomials in 1 L i : i [k]} and in 1 L i : i [k]}, respetively. For example, η 3 = 3 1L L 1 1 L L 1 2 1L L 2 H 6 V2,m µ; Z. Finally, let a=ev 1 γ P n H2 Vk,m µ; Z, where γ P n P n denotes the tautologial line bundle. We next desribe a generalization of the splitting 2.2 whih is used in omputations in Setion 3. If T =S 2,I,[N] M 0 ;j,d is a bubble type, let T = S 2,Ī,[N] M 0 ;j [N] M 0,d Ī, where Ī = I i I Î : d i=0 }, M0 = M 0 M i T. Note that if T is semiprimitive, T is basi. Furthermore, U T µ = M Hi T M i T U T µ, 2.4 i I Ī Ū T µ = i I Ī i I Ī M Hi T M i T Ū T µ, 2.5 where M Hi T M i T denotes the main stratum of M Hi T M i T. If i I Ī, by definition, the bundle L i T ŪT µ is the pullbak by the projetion map of the bundle Lˆ0 T 0 i M Hi T M i T = ŪT 0, where T 0 i = S 2,H i T +M i T, ˆ0};ˆ0,0. i We all the latter bundle the tautologial line bundle over M Hi T M i T. This is the universal tangent line at the marked point ˆ0 M Hi T M i T. The deomposition 2.4 for the bubble T M 0 of Figure 3 13

14 l 1 l M 2 ˆ1,l 1,l 2 } k,l k 1,l 2 Figure 5: An Example of the Deomposition 2.4 is illustrated in Figure 5. Finally, if X is any spae, F X is a normed vetor bundle, and δ: X R is any funtion, let F δ = b,v F : v b < δb }. Similarly, if Ω is a subset of F, let Ω δ = F δ Ω. If υ=b,v F, denote by b υ the image of υ under the bundle projetion map, i.e. b in this ase. 2.3 A Strutural Desription We now desribe the struture of the spaes V k,m µ and the behavior of ertain bundle setions over V k,m µ near the boundary strata. If b= S 2,M,I;x,j,y,u B T and k I, let D T,k b = du k e. If T is a basi bubble type, the maps D T,k with T T and k I Î indue a ontinuous setion of ev TP n over Ū0 T and a ontinuous setion of the bundle L k T ev TP n over Ū T, desribed by D T,k [b, k ] = k D T,k b, if b U 0 T, k C. Proposition 2.7 Suppose p > 2, n 2, d 1, N 1, µ = µ 1,...,µ N is an N-tuple of proper subvarieties of P n in general position, suh that l=n odim C µ odim C µ l N = dn + 1 1, l=1 and M 0 is a subset of [N]. If T =S 2,[N] M 0,Ĩ; j, d is a basi bubble type suh that di =d, the spae Ū T µ is an ms-manifold of real dimension 2 n+1 2 Ĩ M 0 and L k T for k Ĩ and ev TP n are ms-bundles over Ū T µ. If T =S 2,[N] M 0,I;j,d< T, there exist δ,c C U T µ; R + and a homeomorphism γ µ T : FT δ Ū T µ, onto an open neighborhood of U T µ in Ū T µ suh that γ µ T U T µ is the identity, γ µ T FT δ F T is ontained in Ū T µ, and γ µ T F T δ is an orientation-preserving diffeomorphism onto an open 14

15 h 3 h 1 h 2 l D T,k γ µ T υ = D T,h1 +ε T,h1 υ } υ h1 + D T,h3 +ε T,h3 υ } υ h2 υ h3 k Figure 6: An Example of the Estimate of Proposition 2.7 subset of U T µ. Furthermore, for all k Ĩ, with appropriate identifiations, D T,k γ µ T υ α T,k ρt υ Cbυ υ 1 p ρ T υ υ FT δ, where ρ T υ = υ h h χt FT L h T L ι h T ; υ h = υ i ; ι h I Î, h D ιh T ; h χt α T,k υh h χt = and I k I is the rooted tree ontaining k. h I k χt i Î,h D i T D T,h υ h, Figure 6 illustrates the analyti estimate of Proposition 2.7 in a ase when Ĩ = k} is a singleelement set. Note that, while the stratum U T µ of Figure 6 has odimension three in Ū T µ, the setion D T,k depends only on two parameters of the normal bundle, υ h1 and υ h2 υ h3, at least up to negligible terms. Suh bubble types T will always be hollow in the sense of Definition 2.3 and will not effet our omputations. Proposition 2.7 is a speial ase of Theorem 2.8 in [Z2]; see also the remark following the theorem. The dimension of Ū T µ is obtained as follows: 1 2 dim Ū T µ = dim C U T µ = di n+1 + n 2 Ĩ 1n odim C µ + M 0 i I = n Ĩ M 0. The analyti estimate on D T,k is ruial for the implementation of the topologial tools of Subsetion 2.1 in Subsetion 3.1. If T is semiprimitive, the bundle FT = FT and the setion α T = α T ρ T extend over ŪT µ via the deomposition 2.5. In terms of the notions of Subsetion 2.1, FT, FT F T,γ µ T is a normal-bundle model for U T µ Ū T µ. This normal-bundle model admits a losure if T is semiprimitive. Note that FT is not usually the normal bundle of Ū T µ in Ū T µ if both spaes are viewed as algebrai staks; see [P2]. Proposition 2.7 implies only that the restritions to U T µ of FT and of the normal bundle of ŪT µ in Ū T µ are isomorphi as topologial vetor bundles. For any k,m Z, we define bundle E k,m V k,m µ and homomorphism α k,m : E k,m ev TP n over V k,m µ by E k,m Ū T µ = L i T, αk,m υi i Ĩ = D T,i υ i, i Ĩ i Ĩ 15

16 whenever T =S 2,[N] M 0,Ĩ; j, d is a basi bubble type suh that di =d, Ĩ =k, and M 0 =m. The following lemma will be used in Setion 3. Lemma 2.8 Suppose n 2, d 1, N 1, and µ = µ 1,...,µ N is an N-tuple of proper subvarieties of P n in general position suh that odim C µ = dn+1 1. If T =S 2,[N] M 0,I;j,d is a bubble type suh that U T µ V k,m µ, the restrition of α k,m to the subbundle ET L i T E k,m is nondegenerate over U T µ. i χt Î Proof: The linear map α k,m has full rank on ET over U T µ if and only if the setion αk,m ET } Γ PET U T µ;γ ET ev TP n has no zeros. Note that dim C PET U T µ dim C V k µ + k 1 = n k < n. Thus, it is enough to show that α k,m ET } is transversal to the zero set in PET U T µ if the onstraints µ are in general position. This last fat is immediate from Lemma 2.9. Lemma 2.9 If u: S 2 P n is a holomorphi map of positive degree and e T S 2 is a nonzero vetor, the linear maps are onto. H 0 S 2 ;u TP n T u P n, ξ ξ, ξ H 0 S 2 ;u TP n : ξ =0 } T u P n, ξ e ξ, This lemma is well-known; see Corollary 6.3 in [Z2] for example. 3 Computations 3.1 Summary and Motivation In this setion, we prove Proposition 3.1 Suppose n 2, d 1, and µ=µ 1,...,µ N is an N-tuple of proper subvarieties of P n in general position suh that l=n odim C µ odim C µ l N = dn l=1 Then the number of degree-d genus-one urves that have a fixed generi omplex struture on the normalization and pass through the onstraints µ is given by CR 1 µ = n 1,d µ = 1 2 RT1,d µ 1 ;µ 2,...,µ N CR 1 µ, where 2k n+1 1 k 1 k 1! k=1 n+1 2k l=0 n+1 a l η [ Vk n+1 2k l, µ ]. l 16

17 Proposition 3.1 follows from Proposition 3.2 and Corollaries 3.6 and We use the topologial tools of Subsetion 2.1 and the analyti estimate of Proposition 2.7 to obtain the first orollary in Subsetion 3.2. The derivation of Corollary 3.10 in Subsetion 3.3 is essentially ombinatoris. Proposition 3.2 Suppose n 2, d 1, and µ=µ 1,...,µ N is an N-tuple of proper subvarieties of P n in general position suh that odim C µ = dn+1 1. Then the number of degree-d genusone urves that have a fixed generi omplex struture on the normalization and pass through the onstraints µ is given by n 1,d µ = 1 2 RT1,d µ 1 ;µ 2,...,µ N CR 1 µ, where CR 1 µ = Nα 1,0, i.e. CR 1 µ is the number of zeros of the affine map ψ α1,0, ν : E 1,0 =L 1 ev TP n, ψ α1,0, νυ = ν υ + α 1,0 υ, over V 1 µ for a generi setion ν Γ V1 µ;ev TP n. Proposition 3.2 is basially the main result of the analyti part of [I]. The exat statement is not made in [I], but it an be dedued from the arguments in [I] by omparing with the methods of [Z2]. The general meaning of Proposition 3.2 is the following. The number RT 1,d µ 1 ;µ 2,...,µ N an be viewed as the euler lass of a bundle Γ 0,1 over a losure C of the spae C of smooth maps from a fixed ellipti urve that pass through the onstraints µ 1,...,µ N ; see [LT]. Then, 2n 1,d µ = 1 0 C = RT1,d µ 1 ;µ 2,...,µ N C MT µ, 3.1 where M T µ } are omplex finite-dimensional, usually non-ompat, manifolds that stratify 1 0 C C. Equation 3.1 is an infinite-dimensional analogue of 2 of Proposition 2.4. In the finite-dimensional ase, omputation of a ontribution to the euler lass from an s-regular stratum Z of the zero set of setion s redues to ounting the zeros of a polynomial map between finite-rank vetor bundles over Z, unless Z is s-hollow. The goal in the infinite-dimensional ase under onsideration is a redution to the same problem and involves an adoption of the obstrution-bundle idea of [T]. It turns out that C MT µ = 0 for all but one stratum M T µ of 1 0 C C. The number CR 1 µ desribed by Proposition 3.2 is the ontribution C MT µ from the only stratum M T µ of 1 0 C C that does ontribute to the euler lass RT 1,d µ 1 ;µ 2,...,µ N of Γ 0,1. As Subsetion 2.1 suggests, the omputation of Nα 1,0 may require going through a possibly large tree of steps. We onstrut this tree in the next subsetion. However, as a motivation, in the rest of this subsetion, we go through the initial steps of this omputation, without introduing any additional ombinatorial notation. In fat, there are no more steps to go through if n=2 and all the onstraints are points or if n=3 and all the onstraints are points or lines. Sine the domain of the linear map α 1,0 is a line bundle, α 1,0 =α 1,0. Thus, by Lemma 2.5, N α 1,0 = ev TP n E 1,0 1, [ V1,0 µ ] C α 1 1,0 0 α 1,0,

18 where α 1,0 : E 1,0 ev TP n /C ν 0 denotes the omposition of the linear map α 1,0 : E 1,0 ev TP n with the quotient projetion map π 0. As in Lemma 2.5, ν 0 Γ V 1,0 µ;ev TP n is a generi nonvanishing setion. Suh a setion exists, sine the dimension of V 1,0 µ is n 1. We denote the quotient bundle ev TP n /C ν 0 by O 1. Let T =S 2,[N], ˆ0};ˆ0,d. By definition, V 1,0 µ = U T µ. Suppose T = S 2,[N],I;j,d T is a bubble type, i.e. U T µ is one of the spaes of stable maps that stratify V 1,0 µ. If dˆ0 0, by Lemma 2.8, α 1 1,0 0 U T µ =. On the other hand, if dˆ0 =0, by definition, α 1,0 vanishes on U T µ. Thus, α 1 1,0 0 = [T ] U T µ, 3.3 where the union is taken over all equivalene lasses of bubble types T S 2,[N],I;j,d < T suh that dˆ0 =0. By Proposition 2.7 and Lemma 2.8, the deomposition 3.3 satisfies the requirements of 2 of Proposition 2.4, if ν 0 is generi. Indeed, by Proposition 2.7, α 1,0 γ µ T υ α T,ˆ0 ρ T υ Cb υ υ 1 p ρt υ υ FT δ, where 3.4 ρ T υ= υ h h χt FT L ˆ0 T L h T ; υ h = υ i ; α T,ˆ0 υh h χt = D T,h υ h. h χt i Î,h i h χt By Lemma 2.8 and the deomposition 2.4, the linear map α T,ˆ0 : FT HomE 1,0,ev TP n is injetive on every fiber of FT. If the setion ν 0 is generi, the same is true of the linear map π 0 α T,ˆ0 : FT HomE 1,0, O 1, π0 α T,ˆ0 } υ} υ = π 0 αt,ˆ0 υ}υ, 3.5 as an been seen from a dimension ount. Thus, 3.4 implies that there exists C C U T µ; R suh that α 1,0γ µ T υ π 0 α }ρ T,ˆ0 T υ Cb υ υ 1 p π0 α }ρ T,ˆ0 T υ υ FT δ. 3.6 By definition, the ranks of FT and FT are Î and χt, respetively, while χt Î. Thus, by Definition 2.1, U T µ is α 1,0 -hollow if χt Î. In suh a ase, by Proposition 2.4, C UT µα 1,0 =0. On the other hand, if χt =Î, i.e. T is a semiprimitive bubble type, ρ T is the identity map, and thus π 0 α is the resolvent of T,ˆ0 α 1,0 near U T µ. By Proposition 2.7, C UT µα 1,0 = N π 0 α T,ˆ0, where π0 α T,ˆ0 ΓŪT µ;homft,e 1,0 O 1, 3.7 provided π 0 α T,ˆ0 is a regular linear map. By a slight abuse of notation, we now denote by π 0 α T,ˆ0 the extension of the linear map over U T µ defined in 3.5 to ŪT µ. The existene of an extension 18

19 ˆ0 l M 0,4 V 2,1 µ Figure 7: A Boundary Stratum that Contributes to C α 1 1,0 0α 1,0 and Two That Do Not ˆ0 l ˆ0 follows from the deompositions 2.4 and 2.5. With respet to the latter deomposition, FT π1γ ˆ0 π 2 L h T MχT Mˆ0 T Ū T µ, E 1,0 O 1 π1γ ˆ0 π 2O 1 ; h χt π0 α T,ˆ0 } uˆ0 υ h h χt } υˆ0 = uˆ0 υˆ0 π 0 D T,h υ h, h χt where γˆ0 M χt Mˆ0 T denotes the universal tangent bundle for the marked point ˆ0. Thus, summing 3.7 over all equivalene lasses of semiprimitive bubble types T < T, we obtain C α 1 1,0 0α 1,0 = N π 0 α T,ˆ0 = N α 1;k,m, where [T ] k,m>1,0 α 1;k,m Γ M 0,k+m+1 V k,m µ;homγ ˆ0 E k,m;γ ˆ0 O 1, α1;k,m uˆ0 υ} υˆ0 = uˆ0 υˆ0 π 0α k,m υ. Above k,m>1,0 means that k 1, m 0, and at least one of the inequalities is strit. In the proess of omputing the numbers Nα 1;k,m, we will show that π 0 α T,ˆ0 is indeed a regular linear map, as needed. In Figure 7, we give examples of one type of boundary strata U T µ that ontributes to C α 1 1,0 0α 1,0 and of two that do not. As before, eah disk denotes a sphere, and we represent the entire spae U T µ by drawing the domain of an element of U T µ. We shade the omponents of the domain on whih every map in U T µ is nononstant and leave blank the omponents on whih every map in U T µ is onstant. In this figure, we also illustrate the splitting and the summation of over all equivalene lasses of semiprimitive bubble types used in the previous paragraph. In short, the strata U T µ that ontribute to C α 1 1,0 0α 1,0 onsist of the stable maps that are onstant on the priniple omponent, i.e. the one ontaining the speial marked point ˆ0, have only one level of bubbles, i.e. all the non-priniple omponents are attahed diretly to the priniple omponent, and the maps are nononstant on eah of the bubbles. We next apply the topologial method of Subsetion 2.1 to ounting the zeros of an affine map with the linear term α 1;k,m. By Lemma 2.5, N α 1;k,m = γ ˆ0 E k,m γ ˆ0 O 1 1, [ M0,k+m+1 V k,m µ ] C α 1 1;k,m 0 α 1;k,m,

20 where denotes the omposition of the linear map with the quotient projetion map π 1. As before, is a generi non-vanishing setion. We put α 1;k,m : γ ˆ0 γ E k,m γ ˆ0 π PE k,m O 1 / C ν1 α 1;k,m : γ ˆ0 γ E k,m γ ˆ0 π PE k,m O 1 ν 1 Γ M 0,k+m+1 PE k,m ;γ ˆ0 π PE k,m O 1 O 2 = γˆ0 γ ˆ0 π PE k,m O 1 / C ν1 π PE k,m ev TP n /C ν 0 / ν1 γˆ0. Let T = S 2,[N] [M 0 ],Ĩ; j, d be a bubble type suh that M 0 =m, Ĩ =k, d i > 0, and di = d, i.e. Ū T µ is one of the omponents of the spae V k,m µ. Suppose T = S 2,[N],I;j,d T is a bubble type, i.e. U T µ is one of the spaes of stable maps that stratify Ū T µ. By Lemma 2.5, α 1 1;k,m 0 M 0,k+m+1 PE k,m UT µ = b;[υ i i Ĩ ] : υ i =0 if d i 0 }. 3.9 Of ourse, the set on the right-hand side of 3.9 is empty if d i 0 for all i Ĩ. From 3.9, we onlude that α 1 1;k,m 0 = b;[υi i Ĩ ] : υ i =0 if d i 0 }, 3.10 [T ]<[ T ] where the union is taken over all equivalene lasses of bubble types T and T as above. One might think that the deomposition 3.10 is the analogue of 3.3 in this ase, i.e. eah spae on the right-hand side of 3.10 is either α 1;k,m -hollow or α 1;k,m-regular. In general, this is not the ase, and we need to deompose eah spae on the right-hand side of 3.10 into the subspaes based on whih of the omponent elements υ i are not zero. If T and T are bubble types as above and J is a subset of Ĩ, we set ZT J M 0,k+m+1 PET J J JPET J, where ET J = i J L i T U T µ. Let Ĩ0T =i Ĩ : d i =0}. This is the subset of the priniple omponents on whih every k-tuple of stable maps in U T µ is onstant. By 3.10, α 1 1;k,m 0 = [T ]<[ T ] J Ĩ0T Z J T We will show that the set ZT J is α 1;k,m -regular if T is a semiprimitive bubble type and J =Ĩ0T. Otherwise, ZT J is α 1;k,m-hollow and thus does not ontribute to 1 C α 0 α 1;k,m. Figure 8 shows 1;k,m one of a typial main stratum U T µ of V 2,m µ, in a ase when Ĩ = i 1,i 2 } is a two-element set, and two strata U T µ suh that T < T is a semiprimitive bubble type. 20

21 h 1 h 2 h 1 h 2 ˆ0 i 1 i 2 ˆ0 i 1 i 2 Î =h 1,h 2 } Ĩ 0 T =i 1 } χt =i 2,h 1,h 2 } ˆ0 h 3 l i 1 i 2 Î =h 1,h 2,h 3 } Ĩ 0 T =i 1,i 2 } χt =h 1,h 2,h 3 } Figure 8: A Stratum U T µ, s.t. Ĩ =2, and Two Strata U T µ, s.t. T is Semiprimitive The map γ µ T of Proposition 2.7 indues an orientation-preserving homeomorphism γj T neighborhoods of ZT J in between open N Z J T FT γ ET J ET Ĩ0T J ET Ĩ Ĩ0T Z J T and in M 0;k+m+1 PE k,m. The estimate of Proposition 2.7 implies that for some δ,c C Z J T ; R+, αk,m γt J b;υ,u α J T ρ J T b;υ,u 1 Cb υ p ρ J T b;υ,u υ,u N Z J T,δ, 3.12 where ρ J T : N Z J T Ñ ZJ T Ñ h ZT J, Ñ h ZT J = L ι h T L h T, if h Î, ι h J; γ L ET J h T, otherwise; h χt ρ T ;h υ, if h Î, ι h J; ρ J T ;h υ,u = u ιh ρ T ;h υ, if h Î, ι h J; u h, if h Ĩ Ĩ0T ; α J T Γ ZT J ;Hom Ñ ZT J,Homγ ET J,ev TP n α J T υ h h χt } υi i J = D T,h υ h υ i + D T,h υ h υ ev TP n. i J h χt D i T i Ĩ J h χt D i T Above ρ T ;h denotes the hth omponent of ρ T, i.e. υ h in the notation of Proposition 2.7, and ĩ h Ĩ is defined by ĩ h h whenever h I. By Lemma 2.8 and the deomposition 2.4, the linear map α J T : Ñ Z J T Hom γ ET J,ev TP n is injetive on every fiber of Ñ ZT J. If the setions ν 0 and ν 1 are generi, the same is true of the linear map π 1 π 0 α J T : Ñ ZT J Hom γ ˆ0 γ ET J,γ ˆ0 O 2 Hom γet J, O 2 π1 π 0 α J T } υ} υ = π 1 π 0 α J T υ } υ, 3.13 as an been seen from a dimension ount. Thus, by 3.12, α 1;k,m γj T υ π 1 π 0 α J T }ρj T υ Cbυ υ 1 p π1 π 0 α J T }ρj T υ υ N Z J T,δ By definition, the ranks of N Z J T and Ñ ZJ T are Î + Ĩ J and χt, respetively, while χt Î = Ĩ Ĩ0T. 21

22 Thus, if Î χt, i.e. T is not semiprimitive, or J Ĩ0T, the rank of N Z J T is less than the rank of Ñ ZT J and thus ZJ T is α 1;k,m -hollow. On the other hand, if Î χt = and J =Ĩ0T, ρ J T is the identity map, and thus π 1 π 0 α J T is the resolvent of α 1;k,m near ZJ T. By Proposition 2.7, C Z J α T 1;k,m = N π 1 π 0 α J T, where π1 π 0 α J T Γ Z T J ;HomN Z J Z,γ ET O J As before, we now denote by π 1 π 0 α J T the natural extension of the map defined in 3.13 over ZJ T. While we an proeed by omputing the numbers N π 1 π 0 α J T, where T is a semiprimitive bubble type and J =Ĩ0T, we simplify the omputation a little by replaing the linear map π 1 π 0 α J T by another linear map α 2;T, suh that N π 1 π 0 α J T = N α2;t 3.16 and π 1 π 0 α J T is a regular linear map if and only if α 2;T is. With respet to the deomposition 2.5, N Z J T γ ˆ0;i L h T γ ET L J h T, ET J = γˆ0;i M 2;T M Hi T M i T, i J ι h =i h Ĩ J i J i J π1 π 0 α J } T } υ h h χt υi i J = π1 π 0 D T,h υ h υ i + D T,i υ i υ, i J ι h =1 i Ĩ J where γˆ0;i M H i T M i T is the tautologial line bundle. We define the linear map α 2;T by Let ρ: γ ET J E T N Z J T α 2;T Γ M 0,k+m+1 PET J Ū T µ;hom γ ET E T,γ J ET O J 2, α2;t u υh h Ī} υ = u υ π1 π 0 D T,h υ h h Ī be the vetor-bundle map defined by u π ιh υ h, if h ρu υ h = Î; u υ h, if h Ĩ J, where π i : ET J L i T is the projetion map. The map ρ is an isomorphism over the dense open subset Z T of ZT and α 2;T = π 1 π 0 α J T ρ. Thus, 3.16 holds by definition of Nα; see Subsetion 2.1. Summing 3.15 over all equivalene lasses of bubble types T < T of the appropriate form and using 3.16, we onlude that 1 C α 0 α 1;k,m = Nα 2;T = 1;k,m [T ] σ N α σ, where α σ Γ M k+m+1 PF σ V σ ;Hom γ F σ E σ,γ F σ O 2, ασ u υ } υ = u υ π 1 π 0 α συ. This sum is taken over all tuples σ=2;k 2,m 2 ;φ, where k 2,m 2 >k,m and φ speifies a splitting of the set [k 2 ] into k-disjoint subsets and an assignment of m 2 m of the elements of the set [m 2 ] to these subsets. For suh a tuple σ, we put V σ = V k2,m 2 µ; E σ =E k2,m 2 ; α σ =α k 2,m 2 ; F σ = γ σ;i M σ i Imφ i Imφ M i φ 1 i. 22

23 For the purposes of the last line, we view φ as a map from [k 2 ] [k] and a subset of [m 2 ] to [k], and γ σ;i M i φ 1 i denotes the tautologial line bundle. 3.2 A Tree of Chern Classes In this subsetion, we prove Corollary 3.6 by setting up a possibly large, but finite, tree. If eah node of the tree is assigned the hern lass that appears in the statement of Lemma 3.3, the sum of these hern lasses, ounted with a sign dependent on the distane to the root, is the number of Corollary 3.6. The reader is referred to the previous subsetion for a more expliit desription of the first two levels of the tree and for the proof of Lemma 3.3 in the orresponding ases. The proof of Lemma 3.3 in general is nearly the same as the one given for the seond-level nodes in the previous subsetion. Eah node in the tree is a tuple σ=r;k,m;φ, where r 0 is the distane to the root σ 0 =0;1,0;, k 1, and m 0. The tree satisfies the following properties. If r > 0 and σ =r 1;k,m ;φ is the node from whih σ is diretly desendent, we require that k,m < k,m. Furthermore, φ speifies a splitting of the set [k] into k -disjoint subsets and an assignment of m m of the elements of the set [m] to these subsets. This desription indutively onstruts an infinite tree. However, we will need to onsider only the nodes σ=r;k,m;φ with 2k+m n+1. We will write σ σ to indiate that σ is diretly desendent from σ. For eah node in the above tree, we now define a linear map between vetor bundles over an msmanifold. If σ=r;k,m;φ, let σ s =s;k s,m s ;φ s : 0 s r} be the sequene of nodes suh that σ r =σ and σ s σ s 1 for all s>0. Put V σ = V k,m µ, E σ =E k,m V σ, α σ = α k,m, X σ = Y σ V σ, X σ,s = Y σ,s V σ, where Y σ = Y σ,r, Y σ,0 = pt}, Y σ,s = PF σs Y σ,s 1 if s>0, M σ = M i φ 1 i, F σ = γ σ;i M σ. i Im φ i Im φ For the purposes of the last line above, we view φ as a map from [k] [k ] and a subset of [m] to [k ] in the notation of the previous paragraph. Then, γ σ;i M i φ 1 i is the tautologial line bundle; see Subsetion 2.2. Denote by γ Fσ,0 the trivial line bundle over Y σ,0. Let O σ = O σ,r, O σ,0 = ev TP n, O σ,s = O σ,s 1 / Im νσ,s 1 if s>0, where ν σ,s Γ X σ,s ;Homγ Fσs, O σ,s is a generi setion. Sine k s 1 k s, m s 1 m s, and one of the inequalities is strit, 1 2 dim X σ,s 1 2 dim X σ = n+1 2k m s=r + Imφs 1 = n k r < rk Oσ,0 r. Thus, we see indutively that eah bundle O σ,s is well-defined and a generi setion ν σ,s of Homγ Fσ,s, O σ,s does not vanish. Let π σ : ev TP n O σ be the projetion map. We define α σ Γ X σ ;Homγ F σ E σ ;γ F σ O σ, by α σ u υ } υ = u υ π σ α σ υ O σ. Note that α σ0 =α 1,0. s=1 23

24 Lemma 3.3 For every node σ, N α σ = γ O Fσ σ γ E 1, [ ] Fσ σ Xσ σ σ N ασ. Remark: For a dense open subset of tuples ν σ,s }, the orresponding linear map α σ onstruted above is regular and Nα σ is independent of the hoie of ν σ,s }. What we need is that for every bubble type T suh that U T µ V kr,m r µ the intersetion of the image of the linear map α T Γ Y σ U T µ;hom L i T,ev TP n, α T υ = D T,i υ i, with the subbundle i χt Im ν σ,0... Im ν σ,r 1 O σ,0 = ev TP n i χt is 0}. The fat that this ondition is satisfied for a dense open subset of tuples ν σ,s } follows by a dimension ount as above, along with an argument similar to the proof of Lemma 3.10 in [Z2]. Proof of Lemma 3.3: 1 By Lemma 2.5, N α σ = γ O Fσ σ γ E 1, [ Fσ σ Xσ ] C α 1 σ 0 α σ Let σ =r ;k,m ;φ. By Lemma 2.8, α 1 σ 0 is the union of the sets ZT J Y σ PET J L i T U T µ, J JPET J, where ET J = i J taken over non-basi bubble types T = S 2,[N] M 0,I;j,d, with I Î = k, M 0 = m, and di =d, and nonempty subsets J of I Î χt. 2 The map γ µ T of Proposition 2.7 indues an orientation-preserving homeomorphism γj T between open neighborhoods of ZT J in N Z J T FT γ ET J ET I Î χt J ET χt Î Z J T and in Y σ PE σ. Furthermore, the estimate 3.12 holds. Proeeding as in the previous subsetion, we onlude that ZT J is α σ -hollow unless T is semiprimitive and J =I Î χt. Thus, C Z J T α σ = 0 if T is not semiprimitive or J I Î χt If T is semiprimitive and J =I Î χt, we find that C Z J α T σ = N ασ,t if T is semiprimitive and J = I Î χt, 3.20 where α σ,t Γ Y σ PET J Ū T µ;homγ ET J E T,γ ET J O σ,r +1, ET J γ T ;i M σ,t Hi T M i T, O σ i J i JM,r+1 = O σ / Im νσ ασ,t u υ } υ = u υ π σ α k,m υ, k= χt = Ī, m = m + M i T ; i I χt 24