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The local Gromov-Witten invariants of configurations of rational curves Karp, Dagan 2005

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The Local Gromov-Witten Invariants of Configurations of Rational Curves by Dagan Karp M.Sc, Tulane University, 2001 B.Sc, Tulane University, 1999 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in THE FACULTY OF GRADUATE STUDIES Mathematics The University of British Columbia April 2005 © Dagan Karp, 2005 Abstract We compute the local Gromov-Witten invariants of certain configurations of rational curves in a Calabi-Yau threefold. We first transform this from a problem involving local Gromov-Witten invariants to one involving global or ordinary invariants. We do so by expressing the local invariants of a configuration of curves in terms of ordinary Gromov-Witten invariants of a blowup of C P 3 at points. The Gromov-Witten invariants of a blowup of C P 3 along points have a symmetry, which arises from the geometry of the Cremona transformation, and transforms some difficult to compute invariants into others that are less difficult or already known. This symmetry is then used to compute the global invariants. ii Contents Abstract i i Contents i i i Acknowledgements v Dedication v i 1 Introduction 1 1.1 Motivation 1 1.2 Background and Notation 2 1.2.1 Gromov-Witten invariants 2 1.2.2 Local Gromov-Witten invariants 3 1.2.3 Trees of rational curves 4 1.2.4 The blowup of C P 3 at points 6 1.3 Main results 7 1.3.1 The closed topological vertex 8 1.3.2 The minimal trivalent configuration 9 1.3.3 A chain of rational curves 10 1.4 Brief overview 10 2 Configurations of rational curves 12 2.1 A geometric construction . . . 12 2.2 Intersection products and normal bundles 14 3 From local to global invariants 20 3.1 The closed topological vertex 20 3.2 The minimal trivalent configuration 23 3.3 A chain of rational curves 28 iii 4 Properties of the invariants of the blowup of P 3 at points 32 4.1 A vanishing Lemma 32 4.2 The geometry of the Cremona transformation 36 5 Proofs of the main theorems 42 5.1 Proof of Theorem 1 42 5.2 Proof of Theorem 2 42 5.3 Proof of Theorem 3 43 A p p e n d i x A C a l a b i - Y a u configurations in blowups of P 3 45 A . l Another geometric construction 45 A.2 Intersection products and normal bundles 47 A.2.1 The Calabi-Yau condition 47 A.2.2 Normal bundles . 50 Bibl iography 52 iv Acknowledgements First and foremost I thank my adviser, Jim Bryan, for his unsurpassed generosity, unwavering support, unfailing optimism and his unmistakable intuition and style. In addition, I wish to thank Kai Behrend, Jim Carrell, Kalle Karu, Sergiy Koshkin, Melissa Chiu-Chu Liu, Herbert Medina, Maciej Mizerski, Victor Moll, Christoph Miiller, Jim Rogers, Ivelisse Rubio, Jim Voytas and Liam Watson. Also, this work would not have been possible without my supporting family, especially all four of my wonderful parents. Finally, I thank Legier Biederman, the love of my life. DAGAN KARP The University of British Columbia April 2005 To my grandparents, Berna and Gabe Allen, and in loving memory, Sanford and Helen Karp. And to Juan Carlos Fontanive. vi Chapter 1 Introduction 1.1 Motivation The relationship between contemporary physics and mathematics is exciting and deep. String theory, conformal and topological field theories and mirror symmetry have become tremendously important in representation theory, combinatorics, low-dimensional topology, knot theory and symplectic and algebraic geometry. Gromov-Witten theory is an example of a particularly rich interaction be-tween these two disciplines. The subject arose from the study of string theory on Calabi-Yau threefolds. It was noticed that the coefficients of certain correlation functions correspond to the number of holomorphic maps from the worldsheet of the string to the Calabi-Yau threefold. Thus, Gromov-Witten theory was seen to predict answers to questions in enumerative geometry. Of particular significance, for instance, was the prediction of the number of rational curves on the quintic threefold [9]-One main reason that one may be motivated to study the local Gromov-Witten invariants of configuration of rational curves is related to the topological vertex. The Gromov-Witten invariants of any toric Calabi-Yau threefold can in principle be computed by the virtual localization of Graber and Pandharipande [16], which reduces the invariants to Hodge integrals, which can then be computed recursively [12, 20, 24]. However this method is impractical for higher genus invari-ants. A more efficient algorithm to compute the all genus Gromov-Witten invari-ants of any toric Calabi-Yau threefold was proposed by Aganagic, Klemm, Marino and Vafa [1]. The basic ingredients are the topological vertex, which is a generating function for certain open Gromov-Witten invariants, and a gluing algorithm which expresses the Gromov-Witten invariants of any toric Calabi-Yau threefold in terms of the topological vertex. The topological vertex itself is computed using duality be-1 tween Chern-Simons theory and Gromov-Witten theory. This work is conjectural, as neither open Gromov-Witten invariants nor Chern-Simons/Gromov-Witten duality are well defined mathematically. In an attempt to overcome these difficulties, J. L i , K. Liu, C.-C. Liu, and J. Zhou [21] introduce formal relative Gromov-Witten invariants and degeneration as an alternative to open invariants and gluing, and they define an alternate generating function. It is shown that the degeneration formula agrees with the gluing formula. To show that the two algorithms are in complete agreement, it remains to show that the generating functions are equal. In order to prove equality, it is necessary to compute so called one-, two-, and three-partition Hodge integrals. The proof has been completed in case one of the partitions is empty, and the full three-partition case remains open. Consequently, the local Gromov-Witten invariants of configurations of ratio-nal curves, in case the configuration is not contained in a surface in a Calabi-Yau threefold, are conjectured but not proved using the vertex technology. Thus, com-putations of these local invariants provide a check of the conjecture in that case. For example, the closed topological vertex is defined to be a configuration of three IP-'-'s meeting in a single triple point in a Calabi-Yau threefold, with some mild assump-tions about the formal neighborhood. In general it is not contained in a surface and its local invariants were computed by Bryan and Karp [5], which appears as Theo-rem 1 here. Theorem 2 provides another infinite family of potential verifications of the vertex conjecture. In addition to the above motivation, local Gromov-Witten invariants of ra-tional curves are interesting in and of themselves. The local invariants of curves have been studied in [5, 7, 8, 11, 23]. Also, the local invariants of ADE configurations of rational curves were computed by Bryan, Katz and Leung in [6]. Our computations are new and add to this list. 1.2 Background and Notation 1.2.1 Gromov-Witten invariants We now recall the central objects in the Gromov-Witten theory of (local) Calabi-Yau threefolds and establish notation. Let X be a smooth complex projective algebraic variety of dimension three. We may identify H* (X, Z) with H* (X, Z) as rings via Poincare duality, where in-tersection product is dual to cup product. We use the convention that curve classes (and not necessarily curves them-selves) will be denoted by lower case letters, and divisor classes will be denoted by 2 the upper case. For example, we may denote the class of a hyperplane in P 3 by H E # 4(P 3, Z) and the class of a line by h € # 2(P 3, Z). And we may denote a curve by C C X, but we will specify its class by [C] e H*(X,Z). We denote the canonical bundle of X by Kx • We say that X is Calabi- Yau if it has trivial canonical bundle, Kx — Ox, and we say that X is locally Calabi-Yau near the curve C C X, or that C is a locally Calabi-Yau curve in X , if Kx • [C] — 0. Let X be locally Calabi-Yau near C C X and let [C] = B e #2(X, Z). Then, we let Mg(X,B) denote the moduli space of stable genus g maps to X representing 8. It is constructed and shown to be a proper algebraic stack by Fulton and Pandharipande in [13]. L i , Tian, Behrend and Fantechi [2, 3, 22] showed that this moduli space comes equipped with a perfect obstruction theory, which defines a virtual fundamental zero cycle. The genus g Gromov-Witten invariant of X in class 8 is defined to be the degree of this virtual fundamental class; we denote it as follows. <)^:=deg[Mp(X,/?)r = /_ 1 € Q J[Mg{XM™ For a general reference on Gromov-Witten theory, see the excellent books [10, 18]. 1.2.2 Loca l G r o m o v - W i t t e n invariants Let t : Z ^ X be a closed subvariety of the smooth complex projective threefold X, and suppose that X is a local Calabi-Yau threefold near Z. Let 8 be the class of a curve class in Z, and let Mz denote the substack of Mg(X, i*B) consisting of stable maps with image in Z. If Mz is a union of path connected components, then it inherits a virtual 0-cycle (by restricting [M 9(x, t H,^)] v i r to HQ(MZ,Q))- The local Gromov-Witten invariant of Z C X is defined to be the degree of this 0-cycle; we denote it by N9p(Z C X) Note that in general N^(Z C X) depends on a formal neighborhood of Z C X, and in some cases it only depends on the normal bundle. If the neighborhood is understood, we write Np(Z). For a wonderful expository article on the issues surrounding local invariants, see [4]. 3 Figure 1.1: The minimal trivalent tree • • I ' • 1.2.3 Trees of rational curves Let T be a connected graph consisting of vertices V(F) and edges £(T). An edge is specified by an unordered pair of vertices: £ ( r ) c S y m ( V ( r ) , V ( r ) ) Furthermore, we assume throughout that T is a tree. For any vertex v, the valence of v is defined to be the number of distinct edges containing v. We denote the valence of v by | v | = \ { v ' € V(T) : {v,v'} € £(T)}\ We say that the tree V is trivalent if (i) M < 3 for all v € V(T) • (ii) \VQ\ = 3 for some VQ € V(T) Thus there is a unique infinite trivalent tree T 3 ^ such that {3 ii v — VQ 2 otherwise We call it the minimal trivalent tree, and it is shown in Figure 1.1. When there is no chance of confusion, we will also refer to a finite trivalent subtree of the minimal trivalent tree as the minimal trivalent tree. Also, there is a unique infinite tree rf^^ such that \v\ = 3 for all v € V ( r f n a x ) 4 Figure 1.2: The maximal trivalent tree We call ( F ; ^ ) the maximal trivalent tree. It is depicted in Figure 1.2. A configuration of rational curves C is by definition a union along points of non-singular rational curves. j€J Here J is some indexing set. We specify these points below and we specify the local geometry of the intersection points for the configurations of interest in Assumption 1. We say that a configuration of rational curves corresponds to the graph V if there is a one to one correspondence between edges of the graph and irreducible components of C. { C ] C C } ^ f t . ^ ( r ) } Additionally, it is required that there is a one to one correspondence between vertices v with valence greater than one in T and intersection points pv of the corresponding components of C: v G £j f l £k pv £ Cj n Cfc Here the components are necessarily distinct, i.e. j ^ k. The above union is then defined to be the union along these intersection points. C=\JCj {Pv} 5 1.2.4 The blowup of C P 3 at points We briefly review the properties of the blowup of P 3 at points used here for com-pleteness and to set notation. This material can be found in much greater detail in, for instance, [17]. Let X —> P 3 be the blowup of P 3 along M distinct points {pi,... ,PM}- We describe the homology of X. All (co)homology is taken with integer coefficients. Note that we may identify homology and cohomology as rings via Poincare duality, where cup product is dual to intersection product. Let H be the total transform of a hyperplane in P3, and let Ei be the excep-tional divisor over p%. Then Hi(X,Z) has a basis H±(X) — (H, Ei,..., EM) Furthermore, let h G H2(X) be the class of a line in H, and let be the class of a line in Ei. The collection of all such classes form a basis of H2(X). H2(X) = (h,ei,.. .,eM) The intersection ring structure is given as follows. Let pt G HQ(X) denote the class of a point. Two general hyperplanes meet in a line, so H • H = h. A general hyperplane and line intersect in a point, so H • h — pt. Also, a general hyperplane is far from the center of a blowup, so all other products involving H or h vanish. The restriction of Ox(Ei) to Ei = P 2 is the dual of the bundle 0 P2(1), so .Ei • Ei is represented by minus a hyperplane in Ei, i.e. Ei • Ei = —ej, and Ef = (—l)3~1pt = pt ([14]). Furthermore, the centers of the blowups are far away from each other, so all other intersections vanish. In summary, the following are the only non-zero intersection products. H • H = h H-h = pt Ei- Ei = -ei Ei-ei = -pt Also, we point out the important fact that the canonical bundle is easy to describe in this basis. Let Kx denote the canonical bundle of X. Then we have M KX = 4H -2j2Ei i=l Finally, we introduce a notational convenience for the Gromov-Witten in-variants of P 3 blown up at points in a Calabi-Yau class. Any curve class is of the form M P = dh — ^ cnei i=i 6 Ai A2 Figure 1.3: The minimal trivalent configuration for some integers d, ai where d is non-negative. Thus K\ • B = 0 if and only if 2d = Y,T=i ai- I n t h a t case, ( i s determinedby the discrete data {d, a^,..., a^}. Then, we may use the shorthand notation Og,p = (d;ai,...,aM)f. For example, ( )g ,6/i-ei-e2-2e 3-3e5-3e6 = >^ 0, 3, 3)* . Furthermore, the Gromov-Witten invariants of X do not depend on ordering of the points pi, and thus for any permutation a of M points, (d;ai,.., ,ciM)g = (d; aCT(i), • . . , aa{M))^ • 1.3 Main results Our main results determine the local Gromov-Witten invariants of certain config-urations of rational curves C C X inside a local Calabi-Yau threefold. Al l of the configurations considered correspond to connected subtrees of the minimal trivalent tree. The infinite minimal trivalent configuration is depicted in Figure 1.3. Fix a non-negative integer N. As indicated, we label the components of C by N C = ( J Ai U Bi U d , i = i where Ai = Bi = Ci = P1, reflecting the nature of the configuration. 7 We denote the genus-<? local Gromov-Witten invariants of C C X by N^.b (C) where a = ai,... ,aN b = bi,...,bN C = C I , . . . , C J V and bi, Cj is the degree of the map to the component Ai, Bi, Ci respectively. In order for N^.h.c(C) to be well defined, we need to specify the formal neigh-borhood C C X and the local geometry of the intersection points in C. Assumption 1. We assume that the local geometry of C C X is as explicitly con-structed in section 2.1. In particular, C is embedded in X such that the normal bundle of each component of C is given as follows. N A T / X = NB./X = NCi/x 0(- l)©0(- l) ifi = l 0@0(-2) ifl<i<N Additionally, for the case of a\,b\, C\ > 0, we assume the formal neighborhood of the triple point has the geometry of the coordinate axes in C 3 with respect to the local coordinates defined by the normal bundles. We assume all other intersections are nodal singularities. We now state the main results. 1.3.1 The closed topological vertex We now define the closed topological vertex. Let C = Ai U Bi U Ci C X be a locally Calabi-Yau configuration corresponding to a minimal trivalent configu-ration satisfying Assumption 1. Then we call C the closed topological vertex. Theorem 1 (Bryan-Karp). Assume a\,b\,c\ > 0. Then the local invariants of the closed topological vertex are. well defined and given as follows. if {ai, bi, ci} contains two distinct non-zero values, otherwise ^ 1 ; a l i a 1 ( C ) = < ; 0 ; 0 ( C ) -8 Note that N% 0Q(C) is the contribution to the genus g Gromov-Witten in-variant from a single P 1 smoothly embedded by a degree-ai map to a Calabi-Yau threefold with normal bundle 0(—\) © O ( - l ) . These were computed by Faber-Pandharipande [11] to be 1.3.2 The minimal trivalent configuration Theorem 2. Let N C = ( J {Ai U Bi U Q) i=i 6e a configuration of rational curves in a local Calabi- Yau threefold X which corre-sponds to the minimal trivalent tree and satisfies Assumption. 1. Let cii,bi,Ci denote the degree of the map onto the'component Ai,Bi,Ci respectively. Assume a\ = bi — c\ — 1. The local invariants of C are well defined and given as follows. unless a, b, c satisfy 1 = CLl > • • • > <2/v > 0 1 = &i > • • • > bN > 0 1 = ci > • • • > cN > 0 In that case, for any 1 < n, m, I < N, we have • ^ : b : c ( ^ " ,:! I (C) n N—n • m N—m I N—l = *f;l;l(C)-Note that Nf.^C) is the genus g, degree (1,1,1) local invariant of the closed topological vertex. This is a special case of Theorem 1. N 9 ( C ) \B29(2g-l)\ * l i l : l ( C ) " m where B2g is the 2gth Bernoulli number. 9 1.3.3 A chain of rational curves Now consider the case that C = A \ U- • -Li AN is a chain of rational curves; it is shown in Figure 1.4. Note that a chain is of course a subtree of the minimal trivalent tree. Figure 1.4: A chain of rational curves Theorem 3. Let C = AiU---UAN be a chain of rational curves satisfying Assumption 1. Let denote the degree of-the map onto the ith component (here bi = Ci = 0). Assume a\ > 0. Then the local invariants of C are well defined and given as follows. Ni(C) = 0 unless a\ = ai = • • • = a,j = a aj+i = aj+2 = • • • = fljv = 0 for some a > 0 and 1 < j < N. Otherwise Ni(C) = K_a(C) = i V X . . A 0 ( C ) = • • • = N%(C). Note that N£(C) is again the contribution to the genus g Gromov-Witten invariant from a single P 1 smoothly embedded by a degree-a map to a Calabi-Yau threefold with normal bundle C ( - l ) © C ( - l ) . Thus K(0 -1.4 Br ief overview We now give a brief description of the organization of this work and the techniques used to obtain the main results. In chapter 2 we construct the configurations C of rational curves that are the study of this work. They are constructed as locally Calabi-Yau configurations in 10 a blowup space which is deformation equivalent to the blowup of P 3 along distinct points. We next show that the local invariants of the configurations are equal to certain ordinary invariants of the blowup space; this takes place in chapter 3. In order to relate the local invariants to global ones, we first show that inside the blowup space lives a configuration with the correct formal neighborhood. Then, it is left to show that the only contributions to the global invariants in the class of interest come from maps whose image lies in the specified configuration. We do so by using the toric nature of blowup space and using a homological argument. In chapter 4 we study the properties of the Gromov-Witten invariants of the blowup of P 3 along points. This material provides the tools necessary to complete the proofs of the main theorems. In particular, we prove a lemma showing that a large class of invariants of the blowup space vanish. Finally, we make crucial use of the geometry of the Cremona transformation. The Cremona transformation admits a resolution on a space which is the blowup of P 3 along points and lines. The resolved map acts on (co)homology and preserves Gromov-Witten invariants, because it is an isomorphism. This results in a symmetry of the invariants on the resolution space. In order to use this symmetry, we show that for a Calabi-Yau class 8, the invariants of the resolved space descend to invariants of a blowup of P 3 along only points, and not points and lines. This results in a symmetry of the Gromov-Witten invariants of P 3 blown up at points. This study of the Cremona transformation first appeared in [5], was inspired by the beautiful work of Gathmann [15], and is joint work with Jim Bryan. In chapter 5 we use the previous results to first relate the local invariants of the configuration to certain ordinary invariants of the blowup of P 3 at points. Then, we use the tools of chapter 4 to compute the ordinary invariants. As discussed above, the configurations we study are all finite subtrees of the minimal trivalent tree. And what's more, for the most general subtrees we restrict the degree of the invariants. However, in the appendix we construct Calabi-Yau configurations of curves corresponding to any finite subtree of the maximal trivalent tree. We do not compute the invariants of these general configurations or for general degree minimally trivalent configurations because our method fails in those cases. For counter examples, see remark 10. Specifically, it is not the case that the only contributions to the global invariants of the blowup space in the correct class come from maps whose image is the desired configuration. 11 Chapter 2 Configurations of rational curves 2.1 A geometric construction We now construct configurations of rational curves, whose local Gromov-Witten invariants are the study of this work. These configurations correspond to finite sub trees of the minimal trivalent tree. We construct these configurations as subvarieties of a locally Calabi-Yau space X, which is obtained via a sequence of toric blowups ofP 3 : x = x N + 1 7rjv+1> xN nN> • • • X1 —X° = P 3 In fact, Xl+1 will be the blowup of X1 along three points. Our rational curves will be labeled by Ai, Bi, Ci, where 1 < i < N, reflecting the nature of the configuration. Curves and in intermediary spaces will have super-scripts, and their corresponding proper transforms in X will not. The standard torus T = ( C x ) 3 action on P 3 is given by {h,t2,t3) • Oo :x1:x2: x3) i-> (x0 : tixi : t2x2 : £ 3 X 3 ) . There are four T-fixed points in X° := P 3 ; we label them p0 = (1 : 0 : 0 : 0), q0 = (0 : 1 : 0 : 0), r 0 = (0 : 0 : 1 : 0) and s0 = (0 : 0 : 0 : 1). Let A0, B° and C° denote the (unique, T-invariant) line in X° through the two points {po, SQ}, {qo, SQ} and {ro, so}, respectively. Define Xl^XQ to be the blowup of X° at the three points {po,qo,^o}, and let Al,Bl,Cl C X1 be the proper transforms of A°,B° and C°. The exceptional divisor in X 1 over po intersects A1 in a unique fixed point; call it p\ € X1. Similarly, the exceptional 12 Figure 2.1: The T-invariant curves in X2 divisor in X1 also intersects each of B1 and C 1 in unique fixed points; call them q\ and 7*1. Now define X2 ^ X1 to be the blowup of X 1 at the three points {pi,q\,ri}, and let A2,B2,C2 C X2 be the proper transforms of Al,Bl,Cl. The exceptional divisor over p\ contains two T fixed points disjoint from A2. Choose one of them, and call it p2; this choice is arbitrary. Similarly, there are two fixed points in the exceptional divisors above qi, ri disjoint from B2,C2. Choose one in each pair consistent with the choice of p2 and call them q2 and r2. This choice is indicated in Figure 2.1. Let A\ denote the (unique, T invariant) line intersecting A\ and p2. Define B\,C2 analogously. Clearly X2 is deformation equivalent to a blowup of P 3 at six distinct points. The T-invariant curves in X1 are depicted in Figure 2.1, where each edge corre-sponds to a T-invariant curve in X1, and each vertex corresponds to a fixed point. For simplicity, at this time we only label those curves in the configuration. The remaining T-invariant curves will be discussed in Chapter 3. We now define a sequence of blowups beginning with X2. Fix an integer JV > 2. For each 1 < i < N, define to be the blowup of X1 along the three points pi, qi, r^. Let A^x c Xl+l denote the proper transform of Aj for each 1 < j < i. The exceptional divisor in Xl+1 above Pi contains two T fixed points, choose one of them and call it Pi+\. Similarly choose 13 Figure 2.2: The T-invariant curves in X3 qi+i,ri+i, and define Al^\ c Xt+1 to be the line intersecting A\+1 and Pi+i, with BithCt+l defined similarly. The T invariant curves in X2 are shown in Figure 2.2. Terminate this process after obtaining the space XN+1, and define X = XN+1. Finally, define the configuration C c l b y C= | J AjUBjUCf, l<j<N where Aj = A?^ Q - q v - ! . The configuration C is shown in Figure 2.3, along with all other T-invariant curves in X. 2.2 Intersection products and normal bundles We now compute H*(X, Z) and identify the class of the configuration [C] £ H2{X, Z). Al l (co)homology will be taken with integer coefficients. We denote divisors by 14 Figure 2.3: The T-invariant curves in X 15 upper case letters, and curve classes with the lower case. In addition, we decorate homology classes in intermediary spaces with a tilde, and their total transforms in X are undecorated. Let E\,F\,G\ € H±(Xl) denote the exceptional divisors in X 1 —> X° over the points po,qo and ro, and let E\,Fi,G\ € H^(X) denote their total transforms. Continuing, for each 1 < % < N + 1, let Ei, Fi, Gi € H^(Xl) denote the exceptional divisors over the points Pi-i, qi-\,Ti-\ and let Ei, Fi, Gi € H±(X) denote their total transforms. Finally, let H denote the total transform of the hyperplane in X° = P3. The collection of all such classes {H, Ei, Fi,Gi}, where 1 < i < N +1, spans H±(X). Similarly, for each 1 < i < N + 1, let e.i,fi,g~i 6 H2(Xl+1) denote the class of a line in Ei,F,Gi and let ei,fi,gi G denote their total transforms. In addition, let h E H2(X) denote the class of a line in H. Then i ^ P O has a basis given by {/i,ei,/i,0i}. The intersection product ring structure is given as follows. Note that X is deformation equivalent to the blowup of P 3 at 3iV distinct points. Therefore, these H H = h H •h = pt Ei- Ei = -ei Ei- ei = -pt F F = -fi Fi- fi = -pt Gi • 9i = ~9i Gi- 9i = -pt are all of the nonzero intersection products in H*(X). See section 1.2.4 above for details. Lemma 4. In the above basis, the classes of the components of C are given as follows. ifi = l ifl<i<N + l ifi = l ifl<i<N + l /^^  = l if I < i < N + 1. P R O O F : Recall that A\ is the proper transform of a line in P 3 through the two points p\,p2 which are centers of a blowup. Since A\ is in particular a curve, and we have the above basis, it must be the case that N [Ai] = d0h- ^2 dUei + d2,ifi + ditigi i=i h - ei -&2 ei - ei+i h - f i - h fi ~ /i+1 h - 9 i - 92 9i ~9i+l 16 for some integers dij with do non-negative. We study the intersection theory of X in order to determine the coefficients dij. By functoriality of blowups, A\ is isomorphic to a line blown up at the two points pi, P2, which of course is also a line. Since a line and a plane generically meet in a point, and using the above ring structure, we calculate pt = [Ai] • H = d0(pt) + 0 Therefore do = 1. Now, to calculate d^i, dit2, note that the exceptional divisor Ei parameterizes directions in P 3 of intersection with the point pi. Since A\ is the proper transform of a line which intersects (contains) p\ and does so at a unique direction, it must be the case that A i and E\ intersect at a point. We calculate pt = [Ai] • E1 = - d i , i ( - p t ) + 0 Thus di,! — 1- An identical argument also shows that d i ^ = 1. To determine d\j for j > 2, note that A i is the proper transform of a line which is far from the other centers of the blowups. So we calculate 0 = [ A i ] - ^ -= -dij{-pt) and thus d\j = 0 for j > 2. Similarly, we see that d2j = d%j = 0 for j > 2. Thus we have [Ax] = h — e\ — e2-Now inspect Ai for i > 1. Its homology class is of the form N [Ai] = d'0h - J2 d'i,iei + + dki9i 1=1 for some integers d\ • where d'0 is non-negative. Recall that Ai is the proper transform of a line in Ei C X1 containing the point pi, which is subsequently blown up. Since Aj is contained in the total transform of an exceptional divisor, it pairs zero with the total transform of the hyperplane class: 0 = [Ai] • H = d'0(pt) Therefore d'n = 0. 17 Now note that by the functoriality of blowups, Ei is isomorphic to the blowup of P 2 at one point, and the class of a line in Ei is ej and the class of the exceptional line is ej+i- Since Ai meets a generic line in Ei in a point, we calculate pt = [Ai] • ei = -di,i(pt) where the product is taken in Ei. Thus d\j — —1. Furthermore, Ai intersects Ei+i in a point corresponding to the direction of incident with the point pi, as Ei+\ is the exceptional divisor above pi, and so we calculate pt = [Ai] • Ei+i = -d1}i+1(-pt). Therefore d\j = 1. Finally, note that Ai is far from the centers of all other blowups, and therefore all other coefficients must vanish. Thus [Ai] = e, — e^ +i for each i > 2. Note that an identical argument works for Bi and Ci for each i, and therefore the result holds. • We now show that C is a locally Calabi-Yau configuration. Let [/':£—> X] £ Mg(X,B) be a stable map with image in C. We show that X is locally Calabi-Yau near Im(/) counted with multiplicity: Here, /*[£] = B = (ai + bi + ci)/i - aiei - - cipi i v ~ ^ { ( a ^ - a i + l ) e * + l + - hj+l)fi+l + ( C i - C j + l ) f f i + l } • 1=1 where cii,bi,Ci is the degree of the map to the component Ai,Bi,Ci and a/v+i = b/v+i = cyv+i = 0. So, we compute / N Kx • = [AH - 2 ^ A i + Bi + d V i=i ^(ai + b\ + ci)h - aiei - hfi - cxgi ~ /~2 i(ai ~ ai+l)ei+l + (bi ~ hi+l)fi+l + - ci+i)gi+i} i=l =12H • h - 2(2[AL] • [Ai] + 2[Bi) • [Bx] + 2[d] • [d]) = 0 18 We now describe the normal bundles of the components of C in X. These are given as follows. L e m m a 5. NAi/X = NBi/X = NCi/X = O p i ( - l ) © 0 P i ( - l ) ifi = l 0 P i © O P I(-2) ifi>l. PROOF: The equivalence NA./X — N B . / X — N c . / X is easily seen by relabeling points. To calculate NAl/x, let D° C X° be a plane containing the line A0, and let D denote its proper transform in X. Then A\ c D, and NAl/£> is a sub bundle of NAl/x of degree [A\] • [Ai], where the product is taken in D. Note that D is deformation equivalent to the blowup of a plane at two points, ahd [Ai] = h — e\ — e2-Thus, the intersection product in D% is given by [Ai] • [Ai] = (h-e±- e2) • (/i - ei - e2) = -1. The set of planes containing A\ span NAl/x, and the above argument holds for any such plane, so we conclude NAl/x — 0(-l) © 0{-\). Now consider Aj, where i > 1. Note that Aj C Sj. As above, NA./E. is a sub bundle of NA./X of degree [Aj] • [Ai], where the product is taken in Ei. Recall that, by the functoriality of blowups, Ei is the blowup of P 2 at a point, and that a is the class of a line in Ei, and e^ +i is the exceptional divisor. We compute [Ai] • [Ai] = (ei - ei+i) • (ei - ei+1) = 1-1 = 0. We now show that the total degree of the normal bundle is -2, forcing the result to hold. Inspect the defining exact sequence 0 -> TAi -> Tx N A i / x - 0. This implies ci {NAi/x) = ci (TX) • [Ai] - C l (T A i) / J V + l \ = I 4tf - 2 + Fj + Gj J - (ei- e H i ) - 2pt = 0 - 2(pi - pt) - 2pt = -2pt. Thus the total degree of the normal bundle is -2, and so NA./X = 0(a) © 0(h), where a + b = —2. Since we have already shown that (without loss of generality) a = 0, we conclude NAi/x^O®0(-2). • 19 Chapter 3 From local to global invariants In this chapter we show that the local Gromov-Witten invariants of the configura-tions C of rational curves constructed in Chapter 2 are equal to certain global or ordinary Gromov-Witten invariants of a blowup of P 3 along points. 3.1 The closed topological vertex Let C = AU B U C be the closed topological vertex, and let X be the blowup space constructed in section 2.1. Moreover, in this case let N = 1 so that X = X2, and is deformation equivalent to the blowup of P 3 along 2-3 = 6 points. Then the local invariants of the closed topological vertex are equal to certain ordinary invariants of X. Proposition 6 (Bryan-Karp). Let C = AuBlSC be the closed topological vertex, and let X = X1 be the N = 0 blowup space constructed in section 2.1. Let (3 = a(h-ei- e2) + b(h - / i - /2) + c(h - g1 - g2) and assume a,b,c > 0. Then the local invariants of C are equal to the ordinary invariants of X in the class (3. KMC = ( & PROOF: C C X has the correct local geometry by assumption. So in order to prove the proposition, it suffices to show that the only contributions to ( ) ^  are given by maps with image in C. Lemma 7. Let X,C and.(3 be as above. Assume a,b,c> 0. Then Im(/) = C for every stable map [/] 6 Mg(X,(3). 20 h - f i - g i e i Figure 3.1: The T-invariant curves in X P R O O F : We explicitly use the toric nature of the construction. Note that the torus action on X° = P3 lifts to X since the center of each subsequent blowup is T fixed. Thus there is a T action on Mg(X,B), simply by composition. Assume that there exists a stable map [f:E^X]EMg(X,p) such that Im(/) <f_ C. Then there exists a point x G Im(/) such that x £ C. Recall that a one parameter family ip of T is defined to be an element ip G Hom(CX, T) = Z3, where the isomorphism is given by O i , n 2 , n 3 ) i-> ipni,n2,n3 • CX -> T * i—»• (tn\tn\tn3). Moreover, recall that T-invariant subvarieties of X are given precisely by orbit-closures of limit points of one parameter subgroups of T. {T-invariant subvarieties} <—• |T • l i m ^ ( i ) | So, in particular, the limit of the point x under the action of ip is a fixed point. Moreover, since x g1 C and every fixed point is the limit of some one parameter subgroup, there exists ip such that lim ib(t) • x = q, 21 h - f \ - 9 \ 'l -92 / i - f2 h-gi-g2 h-h-h \h — ei — e2 / 1 - / 2 . 7 h - ei - / ] / 'ei — e2 ei - ei Figure 3.2: The remaining possible curves in Im(/") where q is T fixed and q $ C. So, the limit of ip acting on [/] is a stable map /' such that q G Im(/'). It follows that q is in image of all stable maps in the orbit closure of [/']. Thus, there must exist a stable map [/" : S —> X] £ Mg(X:3) such that Im(/") is T invariant and Im(/") cf C. We show that this leads to a contradiction. Let I denote the union of the T invariant curves in X; it is shown in Figure 3.1. Inspect the class /"[£]. Note that the total multiplicity of the e's is a + b + c. Also note that every component of I whose class contains h also contains two —e terms. Therefore the total multiplicity of the terms not containing h must be zero. The same is true for the / and g terms. The curves in I whose class does not contain h either contribute nothing to the multiplicity of the e, /, g's, or they contribute a strictly positive amount. Therefore, the curves whose classes contribute positively to the total multiplicity of the e, /, p's must not be contained in the image of /". Therefore, the curves in class ei,fi,gi are not in Im(/"). The remaining possible curves in the image of /" are shown in Figure 3.2. Recall that Im(/") (L~ C by assumption. Therefore the image of /" is con-tained in the outer components in Figure 3.2. But the image of /" is connected and must and [Im(/")] contains strictly positive multiples of each of e\,f\,gi, since a,b,c > 0. Therefore the image of /" is a union of the outer components compo-nents. Therefore the image of /" is not connected. This contradiction shows that our original assumption is incorrect. Therefore the result holds. 22 3.2 The minimal trivalent configuration We now consider the case when C is a minimal trivalent configuration. Proposition 8. Let N C = [ J (Ai U Bi U d ) i=i be a minimal trivalent configuration of rational curves satisfying Assumptions 1. Let a-i,bi, Ci denote the degree of the map onto the component Ai,Bi, Ci respectively, and let a = (a\,..., ayv), b = (b\,..., 6/v), c = ( c i , . . . , C A T ) . Also, let X continue to denote the blowup space constructed in Chapter 2. Assume a\ = b\ = c\ = 1. Then the local invariants of C are equal to the global Gromov-Witten invariants of X in the class (3, tf£b,c(C) = < > ^ , where (3=(ai + bi+ ci)h - aie x - 61/1 - cigx N ~ {(ai ~ ai+i)ei+i + (Pi - bi+i)fi+i + (ci - ci+1)gi+1} . i=l and a A r + i = b^+i = C J V + I = 0. P R O O F : By assumption, the formal neighborhood of C agrees with the construction in X. Thus, in order to prove the Proposition it suffices to show that the only contributions to ( ) ^ are from maps to C. L e m m a 9. Let X,C and (3 be as above, where we assume that a-i = bi = c\ = 1. Then every stable map [/] € Mg(X,(3) has image C. P R O O F : Assume that there exists a stable map [/ : E —> X] € Mg(X,(3) such that Im(/") <f_ C. Then by the proof of Lemma 7, there exists a stable map [/"] € Mg(X,B) such that Im(/") is T invariant but Im(/") £ C. We show that this is a contradiction. Let F C X denote the union of the T-invariant curves in X; it's shown in Figure 2.3. We study the possible components of F contained in the image of / " . 23 Figure 3.3: The possible curves in Im(/") Suppose that Ax U Bx U C\ C Im(/"). Then /"[£] contains (at least) 3/x. Note that [F] has no. —h terms. Therefore Im(/") does not contain any of the curves h — e\ — fi, h — e\ — gi, h — fi — g\. And furthermore each of A±, B\ and C\ must have multiplicity one. There are no remaining terms that contain —e\, —j\ or — g\. Also, since the image of / " contains precisely one of A\,B\,C\, we conclude that the multiplicity of terms contain positive e\,f\,g\ must be zero. Thus, Im(/") is contained in the configuration shown in Figure 3.3. Now, note that in B the sum of the multiplicities of the e^ 's is -2. This is true of the curve A\ as well. Therefore the total multiplicity of all other e terms must vanish. But all other e terms are of the form — ei+\ or ej. Since, the former 24 contribute nothing to the total multiplicity, we conclude that there are no ej terms in the image of /". Therefore Im(/") must be contained in the configuration shown in Figure 3.4. But Im(/") is connected, and contains h terms. Therefore it can not contain nor be contained in any of the three outer parts of Figure 3.4. Therefore Im(/") C C and thus Im(/") = C. This contradicts our assumption, and therefore at least one of Ai,B1, Ci is not in Im(/"). Without loss of generality, suppose A\ (t Im(/"). Let detf,detg,dftg denote the degree of /" on the components h — e\ — f\, h — e± — g\, h — f\ — g\ respectively. 25 Since A\ is not contained in the image of /" , we must have de,f + de,g > 0. Furthermore, in order for Im(/") to simultaneously be connected and contain —ei terms for i > 1, it must be the case that Im(/") contains two of {ei,ei-e 2,ei eN+1}. Thus de,f + de^g = 3 df,9 = 0 and B i , C i (/L Im(/"). This forces Im(/") to be contained in the configuration shown in Figure 3.5. Again we have that Im(/") is connected and contains —/j, —gj for some i,j > 1. Therefore Im(/") contains at least one of f\ — / 2 , / i — • • • — / J V + I and also at least one of g\ — g2, g\ — gw+i- But the multiplicity of f\ and g\ in /3 is — 1. Therefore de,f, detg > 2. This contradictions shows that our assumption A i (£_ Im(/") is incorrect. Therefore A i C Im(/"). An identical• argument also shows that B i , C i C Im(/"). However we showed above that A\,Bi,Ci <f. Im(/"). This contradiction shows that our original assumption is incorrect. Therefore there does not exist a point x G Im(/") such that x G" C, and Im(/") c C, Therefore Im(/") = C, and the result holds. • Remark 10. Note that this argument does not hold for general ai,6i,ci. For instance, it is a fun exercise to show that there is more than one T invariant config-uration of curves in X in the following classes. /?i =2(h - ex - e2) + (e2 - e3) + 2 0 - / 1 - / 2 ) + ( / 2 - / 3 ) + 2(h - 5i - g2) + ($2 - 93) #2 =4(h - ei - e2) + (e2 - e 3 ) + 2(fc - /1 - / 2 ) + 2(h - g i - g2) + (52 - £3) /33 =40 - ei - e2) + 4(e2 - e3) + 4 0 - /1 - / 2 ) + 4(/ 2 - / 3 ) + 4(/i - gi - 92) + 4(52 - gi)-26 Figure 3.5: The other possibility for curves in Im(/") 27 3.3 A chain of rational curves Next consider the case when C is a chain of rational curves. More precisely, let C = AiU---(JAN+i cX. Since C does not contain any of the curves P>i,Ci, the blowups with centers Pi and qi in the construction of X are extraneous. In order to simplify the argument in this case, consider the space Y = YN+1 *N+1> YN 771) P 3 where the construction of Y follows that of X, without the extraneous blowups. So Yl+1 —• Yl is the blowup of Yl along the point pi, where pi is defined in Section 2.1. Thus, Y is deformation equivalent to the blowup of P 3 at N + 1 points. Since C does not contain the curves Bi, Ci, clearly the formal neighborhood of C in Y agrees with the construction in X. We continue to let Ei be the total transform of the exceptional divisor over Pi, and ei be the class of a line in Ei. Furthermore, we continue to let H denote the pullback of the class of a hyperplane in P3, and h be the class of a line in H. Then, {H, Ei} is. a basis for H$(Y) &n<^ {h, e%} is a basis for H2(Y). The non-zero intersection pairings are given as follows. H • H — h H-h = pt Ei- Ei = -ei Ei-ei = -pt The T-invariant curves in Y are shown together with their homology classes in Figure 3.6 Proposition 11. Let the blowup space Y and the chain of rational curves C — A\ U • • • U AN be as constructed above. Let a = ai,..., a# .where ai is the degree of the map to Ai. Assume a\ > 0. Then the local invariants of C are equal to the global invariants of Y in the class B, where N+l 8 = a\h - aiei - ^ ((H-x - a,i)Ci 2=2 and aN+x — 0. 28 h e i Figure 3.6: The T-invariant curves in Y P R O O F : C C Y has the desired geometry by construction, in order to prove the proposition it suffices to show that the only contributions to the Gromov-Witten invariants of Y in class 8 are from maps to C. Lemma 12. Let Y, C = A \ U • • • U AN and be as above. Then Im(/) = C for any stable map [/] E Mg(Y, /?).. As shown in Proposition 8, we may use the toric nature of Y to construct a stable map [/" : S Y] E Mg(Y,8) such that Im(/") is T invariant, but Im(/") (t C. We show that this leads to a contradiction. We study the class = 8. Note that the multiplicity of the — e± term is the same as that of h: Furthermore, each —e\ occurs along with h, and there are no —h terms. Therefore Im(/") can not contain any terms containing positive e\, nor can it contain any of the curves in class h. Thus, the image of /" is contained in the configuration of curves shown in Figure 3.7. Since a\ > 0, it must be that contains at least one ej term with non-zero multiplicity for i > 1. Also, Im(/") is connected and so we conclude that the image of / must not contain either of the curves of class h — e\ in Figure 3.7. 29 [Ai] = h - e i - e 2 h — ei e2 — e3 e3 — e$ es — eg Figure 3.7: The possible curves in Im(/") Now, note that the total multiplicity of the e terms is —2ai, and that the curve A\ must also have this property. Therefore the sum of all other e terms must be zero. Since the other e terms are of the form a — e$+i or ej, we conclude that Im(/") does not contain any of the curves ej. Thus Im(/") is contained in the configuration depicted in Figure 3.8. [Ai] = h - e i - ei ei — ez ez — e^ e$ — ee &N — e j v + i Figure 3.8: The remaining possible curves in Im(/") 30 However, since Y is connected and contains h, we conclude that Im(/") C C, and therefore Im(/") = C. This contradiction shows that our original assumption is incorrect, and the result holds. • 31 Chapter 4 Properties of the invariants of the blowup of P 3 at points In this chapter we prove results needed for the proofs of the main theorems. 4.1 A vanishing Lemma We use the notation of subsection 1.2.4. Lemma 13. Let X be the blowup o/P 3 at n distinct generic points {x\,..., xn}, and (3 — dh — aiei w ^ = 2~2i=i ai> a n d assume that d > 0 and ai < 0 for some i. Then Mg(X,8) = Hi. Corollary 14. For any n points {xi,..., xn} and X and (3 as above the correspond-ing invariant vanishes; • < & = °-This follows immediately from the deformation invariance of Gromov-Witten invariants and Lemma 13. Lemma 15. Let X be the blowup o/P 3 at n distinct generic points {xi,...,x^} and let (3 = 2~2i=i ~ a^i, with 2d < YA^I^' a n c ^ assume that ai > 0 for all i. Then Proposition 16. For each d > 0, Lemma 13 is equivalent to Lemma 15. To prove Lemma 13 or Lemma 15, it suffices to prove the Lemmas for some particular choice of {x\,... ,xn} or {xi,... ,Xfi}, since if Mg(X, 8) (or Mg(X,B)) is empty for a specific choice of points, then it is empty for the generic choice. 32 P R O O F : We prove that Lemma 1 3 is false if and only if Lemma 1 5 is false. First, without loss of generality, we may reorder the centers of the blowup so that ci\ < • • • < o-m < 0 < a m + i < • • • < an for some 1 < m < n. Choose {x\,... ,xn} so that {x\,... ,xm} are coplanar, as are {xm+i, , xn}. Let D' and D" be the classes of the proper transforms of those planes, respectively, so that D' — H — E\ — • • • — Em D" = H — Em+i — • • • — En. Assume that Lemma 1 3 is false. Then there exists [/] G Mg(X,3). We show that Lemma 1 5 fails by studying the image of /. Im(/) decomposes into several components. Note that 8 • E\ = a\ < 0 . Thus Im(/) has a component(s) contained in Ei. Denote the union of these components by Ci. If Im(/) has any components contained in Ej we denote them by Cj as well. As ej is the class of a line in Ej = P2, it must be that [Cj] = I'jtj for some bj > 0 . Let J C { 1 , . . . ,n} be the indexing set of these components, J = { 1 < j < n\ There exists a component of'Im(/) c Ej}, and let CE be the union, [CE} = /ZCJ. Im(/') decomposes further. As 2d = Y17=i ai-> w e n a v e P ' (D' +' D") — 2d — 2~27=i a i = 0 &nd, because cii < • • • < am < 0 < am+i < • • • < an, 8-D"=(d- aA <(d-JTaA =8-D>, \ i=m+l J \ i = l / SO n d- « i < 0 . i=m+l Thus (8 - [CE]) -D" = d-YJ*i-Y. bJ < °-i = l j>m Therefore there exists a nonempty closed subscheme of Im(/) n D" which does not have components in the exceptional divisors. Denote it by C". 3 3 Let [Im(/)] denote the class of Im(/). Then in summary, we have [Im(/)] = [CE] + [C] + [C'l where •jeJ [C"]=d"h- £ i=m+l n n [C']=(d-d")h-}~2aiei+ }~2 4ei~Y.bier i=l i = m + l j£j Here bj > 0 as noted above. Also d" > 0, as C" is not contained in the exceptional divisors Ej, and d — d" > 0, since Im(/) is connected. For later use, we point out that bi > 0 and as a result 5>>o. jeJ We now show this implies Lemma 15 could not be true. Consider the curve C which consists of the components of Im(/) which are not contained in the exceptional divisors Ej. It is of class n 0 := [Im(/)] - [CE] = [C] + [C] = dh~YJ - J _ bjej. ' i = i jeJ Note that n n 2d = Y<ii <~^rai + ^r bj. i=i i=i jeJ Also 6 • Ei > 0 as C has no component in the exceptional divisors Ej. Thus the pair (X,8) are as in Lemma 15. Clearly there is a stable map g such that [Im(<?)] = [C1] + [C"\ and consequently [g]eM9(X,P), which contradicts Lemma 15. Conversely, assume that Lemma 15 is false, so that there exists [/] € Mg(X, 8). Let X be the blowup of X at € Im(/) where x* is not contained in the excep-tional divisor. Such a point exists as 8 • Ei > 0 and Im(/) is connected. Let a* = — YA=I There is a curve C* C E* in the exceptional divisor over x* of class [C*] = —a*e* such that n Im(/) ^ 0, where e* is the class of a line in E*. Then 34 Define 8 := [C* U Im(/)] = dh — a*e* — a\e\ - • • • — a^e^. Evidently the pair (X, 8) are as in Lemma 1 3 , and as before it is clear that there exists a stable map / with Im(/) = C* and [f)€Mg(X,0), which contradicts Lemma 1 5 . • P R O O F (OF L E M M A 1 3 ) : Suppose there exists [/] e Mg(X,8). Then with the above choice of {x\,..., xn} we have the above decomposition of Im(/). We use induction on d. Suppose d = 1 , then we have 1 — d" > 0 and d" > 0 , which is a contradiction. We now proceed inductively. Consider the curves C and C" defined above. Suppose 2d" > Y:=m+i < ^ d 2(d - d") > Z7=i <* - Etm+i < + by Then n n ^ a1<2d"< £ <-zZhr i=m+l i=m+l j£j This is impossible. Therefore either 2d" < E"=m+i ai o r 2(d ~ d") < E?=i ai ~ Y^i-m+i a'l + Y^jejbj- Therefore by Lemma 1 5 and the inductive hypothesis there is no decomposition of [Im(/)] involving either C' or C", and therefore there is no [f)€Mg(X,8). • Corollary 17. Let X be the blowup o/P3 along points and define 8 = dh—2~27=i aiei where 2d = E i L i ai a n d d > ®- Also define X'^X to be the blowup of X at a generic point p, so that X' is deformation deformation equivalent to the blowup o/P3 at n + 1 distinct points. Let {h1,e[,...,e'n+1} be a basis of H2(X'),. and let 8' = dh' - ]T" = 1  aA-  Then (d;ai,.. .an,0)g = (d;ax,...,an)g P R O O F : This follows from Lemma 1 5 (or equivalently Lemma 1 3 ) . The method of proof used here was used in [ 5 ] to prove what is included as Lemma 2 0 here. This result also follows from the more general results of Hu in [ 1 9 ] , but the proof is easy in this case, so we include it in order for our results to be more self contained. We will show that any [/'] 6 Mg(X',8') has an image disjoint from E'n+1, the exceptional divisor over p. Note that any [/] € Mg(X,8) has an image disjoint from p. It follows that the natural map Mg(X', 8') —> Mg(X, 8) induced by ir is an isomorphism of the moduli space and their virtual classes. Indeed, it will follow that both Mg(X',8') and Mg(X,8) are canonically identified with Mg(X'\E'n+l, 8'). 3 5 Let [/' : E X'} e Mg(X',B'). Suppose Im(/') n E'i+1 + 0. Note that /*[E] • E'i+l = 0. Therefore Im(/') has a component C contained in E[+l. Since d > 0, Im(/') also has a (union of) component(s) C not contained in E'i+1. The classes of these components are given by ' [C] = dh~Y^ ate'i - me'i+l [C'\ = me'i+1 where m > 0. Note that 2d = X)ILi ai- Therefore n 2d < + ' i=i Therefore the component C does not exist by Lemma 15. This contradiction shows that our assumption was incorrect. Therefore Im(/') n E'i+1 = 0, and the corollary is proved. • 4.2 The geometry of the Cremona transformation This section consists essentially of section 5 from Bryan-Karp [5]. In particular, every result in this section is joint with Jim Bryan. It is included so that this document may be self contained. Theorem 18. Let 8 = dh — YA=I a^ei- w ^ = YA=I ai and assume that cu ^ 0 for some i > 4. Then we have the following equality of Gromov-Witten invariants: ( )*/3 = ( )*/?' where 8' = d'h — Y^l=\ a'iei has coefficients given by d' = 3d- 2(ai + a 2 + a 3 + a 4 a'1 = d- (a 2 + a 3 + a 4) a'2 = ' d- (ai + a 3 + a 4) a 3 = d- (ai + a 2 + a 4) a'4 = d- (ai + a 2 + a 3) *6 = a 5 a n = an. 36 In this section, we prove Theorem 18 by studying the geometry of X, the blowup of P 3 at n points, and X, the blowup of X along a certain configuration of six lines. Let X be the blowup of P 3 at n distinct points xi,... ,xn where n > 4. We take the first four points tobe the fixed points of the standard torus action on P 3 and we take the remaining points to be any fixed points of the Cremona transformation: P 3 —* P 3 ( 2 0 : zi : z2 : z3) *->(—: — : — : —). zo zi z2 z3 Remark 19. The case where n — 4 is greater than the number of fixed points is easily handled by including in the blowup locus pairs of points exchanged by the Cremona transformation. However, for notational convenience we will assume that the points xi,...,xn are fixed. Let Ijk, 1 < j < k < 4 be the proper transform of the line through Xj and Xk. Let Tr : X -» X be the blowup of X along the six (disjoint) lines Ijk-X admits an involution r : X —> X which resolves the Cremona transforma-tion. The map r is discussed in more detail by Gathmann in [15], although note that our X has the additional blowups at x§,...,xn whose corresponding excep-tional divisors are simply fixed by r, or possibly exchanged if the points x^...xn include non-trivial orbits (c.f. Remark 19). We briefly describe the divisors and the curves on X and X and their in-tersections. Generally, we denote divisor classes with upper case letters and curve classes with lower case letters. - Classes on X will have a hat, and classes on X will not. The homology groups H4(X; Z) and H2(X;Z) are spanned by the divisor and curve classes respectively: HA(X;Z) = {H,E1,...,En), H2(X;Z) = (h,eu ...,en). Here H is the pullback of the hyperplane in P3, h is the class of the line in H, Ei is the exceptional divisor over X j , and e j is the class of a line in Ei. The intersection pairing on X is given by: H • H — h, Ei • Ei = — ei, H • h = p, Ei-ei = -p 37 where p £ HQ(X; Z) is the class of the point and all other pairings are zero. The homology groups H^(X;1) and H2(X;Z) are also spanned by divisor and curve classes: tf4(X;Z) = (H,Ei,Fjk), H2(X;Z) = (h,eiJjk/, where 1 < i < n and 1 < j < k < 4. Here H is the proper transform of H and h is the generic line in H. Ei is the proper transform of Ei and ii is the class of the generic line in Ej. Fjk is the component of the exceptional divisor of X —> X lying over Ijk, and fjk is the fiber class of rr : Fjk —> Zjfc. Note that Fjfc —> Zjfc is the trivial fibration and the class of the section §jk is given by Sjk = h &j &k + fjk-The intersections are given as follows: Fjk ' Fjk — §jk fjki H H = h, Ei Ei = H Fjk = fjk.i Ej Fjk = fjk H h = P, Ei ei = -P, Fjk ' fjk — Pi where p € HQ(X; Z) is the class of the point and all other intersections are zero. The action of r on divisors is described by Gathmann [15] in section 6. The action of r on the curve classes of X is then easily obtained using Poincare duality and is given as follows: = 3h- (ei + £2 + e 3 + e4 T*e i = 2h- ( e 2 + e 3 + e 4 ) , T*e2 = 2h- ( e i + e 3 + e 4 ) , r * e 3 = 2h- ( e i + e 2 + e 4 ) , T*e 4 = 2h- ( e i + e 2 + e 3 ) , T*e5 - hi T*fjk — Sj'k'i where {/, k'} is defined by the condition {j, k} U {f, k'} = {1,2,3,4}. For a class 8 = dh — Y^i=i w ^ n ^ = J2i=i a*' w e n a v e ~ ^x ' P = ® and so the degree 8 Gromov-Witten invariants have no insertions. Since r is an isomorphism, it preserves the Gromov-Witten invariants of X so in particular, 38 where n T * / ? = d'h - Y^ a'i&i i = l has coefficients d', a[,..., a'n given by the equations of Theorem 1 8 . To prove Theorem 1 8 then, it suffices to prove the following Lemma 20. Let d, a i , . . . , an be such that 2d = Y^7=i a * a n d a i 7 ^ 0 for some i > 4 . Then where 6 = dh — Yli=i ai&i a n d P = dh — Y%=i ai&i-Remark 21. The condition that a? ^  0 for some i > 4 is necessary. For example, 1 = ( ) o , h - e i - e 2 ^ ( )o,k-ei-e2  = °" P R O O F : The lemma follows from the general results of Hu [ 1 9 ] . We warn the reader that the theorems in [ 1 9 ] are incorrect as stated; the above example provides a counterexample. However, the author has informed us that a crucial hypothesis is missing in the main theorems of [ 1 9 ] . Namely, in Hu's notation, he must additionally assume that the class p\(A) is not exceptional. The paper [ 1 9 ] uses the machinery of relative Gromov-Witten invariants and gluing. To make our paper self-contained, we provide below an independent proof of Lemma 2 0 in the case of n = 6 which is what is needed for the closed topological vertex in [ 5 ] . Assume that n = 6 . Without loss of generality we may assume that as ^  0 . We will show that any [/] £ Mg(X,J3) has an image which is disjoint from F = Uj^Fjk, and any [/] £ Mg(X,B) has an image which is disjoint from I = Uj<kljk- It follows that the natural map Mg(X,(3) —> Mg(X,8) induced by ir is an isomorphism of the moduli spaces and their virtual fundamental classes. Indeed, if both Im(/) ( I F = U and Im(/) n I = 0 for all stable maps [/] £ Mg{X,(3) and [/'] £ Mg(X,P), then both Mg(X,(3) and Mg(X,/3) are canonically identified with Mg(X\Fj). _ Let [f : C -> X] e Mg(X,3) and suppose that Im(/) n Ijk / 0 for some j and k. Im(/) gt Ijk since ^ 0 and so MC) = C' + bljk 3 9 where C meets Ijk in a finite set of points (b can be zero here). Let C' be the proper transform of C Since C Pi Ijk ^  0, we have C' • Fjk = m > 0. Therefore we have 6 C' = dh - Y ai&i ~ Hh - ej - e\) - mfjk. Define {f, k'} by the condition {f, k'} U {j, k} = {1,2,3,4} and let Djk = 2H — (Ei H h EQ) — Fjk — Fjiy-Then Djk • C' = -m < 0. However, this contradicts Lemma 22 which states that Djk is nef. Lemma 22. Let 1 < j < k < 4 and define j', k' by the condition {j, k} U {j', k'} = {1,2,3,4}. Then the divisor Djk = 2H — (Ei H h EQ) — Fjk — Fj'k' is nef in X. P R O O F : Let D' and D" be the proper transforms of the planes through {xj,Xk,xs} and {XJ/, Xk>, XQ} respectively. Then D' = H — Ej — Ek — E§ — Fjk D" = H — Eji — E^ — EQ — Fj^' so Djk = D' + D". To see that Djk is nef, it suffices to check that Djk • C > 0 for any curve Cdb'. D' is isomorphic to the blowup of P 2 at three points. Under this identifica-tion, the classes of the line and the three exceptional divisors are h'-=h — fjk, e'j = ij - fjk, e'k — ek-fjk, e'5 = e5. The curve C c D' has class dh! — aje'j — ake'k — 0.5 and since h! — e'b is a nef divisor in D', we have d > a^. 40 The first Chern class of the normal bundle of D' C X is (H - Ej - E k - E 5 - Fjk)2 = -eg = -e'5 and so D' • C = —e'5 • (dh! — cije'j — ake'k — 0565) = —a*, where the intersection product on the right hand side is on D'. Therefore Djk • C = D' • C + D" • C = - a 5 + d > 0. • Thus Im(/) n I = 0 for all [/] e ~Mg(X, 6). We argue in a similar fashion for ~Mg(X,/3). Let [f : C -> X] e ~Mg(X,(3) and suppose that Im(/) C\'Fjk ^  0 for some j and k. Since 8 • Fjk = 0, f*(C) must have a component C" contained in Fjk- We then have P = U(C) = c' + c" where C is non-empty since $ • E§ = a*, > 0. Since C" C Fjk is an effective class in Fjk = IP1 x P1, it is of the form a§jk + bfjk with a, 6 > 0 and a + b > 0. Define as above. Then Djk -(3 = 0 and .Djfc • C" — a + b > 0 and so .Djfc • C" < 0, contradicting the fact that Djk is nef. This proves that Im(/) D F = 0 for all [/] £ Mg(X,$) and Lemma 20 is proved. This then completes the proof of Theorem 18. 41 Chapter 5 Proofs of the main theorems 5.1 Proof of Theorem 1 Let X = X2 be as constructed in section 2.1 and let Y = Y2 be as constructed in section 3.3. By Proposition 6, we have KMCW = ( & where P = a{h-e1- e2) + b(h - /i - h) + c(h - gx - #2)-So, we inspect the invariant () *p = (« + b +C; A> M , &> c' C)J • By Theorem 18, we have (a + fe + c; a, a, fe, fe, c, c) J = (3c — a — fe; c — fe, c — fe, c — a, c — a, c, c)* Thus, by Corollary 14, N^bc(C) — 0 unless a = fe = c. In that case, (3c — a — fe; c — fe, c — fe, c — a, c — a, c, c) J = (a; 0,0,0,0, a, a) J = (a;a,a)J where the last equality follows from Corollary 17. • 5.2 Proof of Theorem 2 Let X = XN+l be as constructed in section 2.1 By Proposition 8, we have *£b :c(c) = ( & 42 where • 3 = 3/x - ei - / i - 9 1 J V + l _ X] ~ + (bi - + (ci - c i +i ) 9 i+i} , i=i where a\ — b\ = c\ = 1 and a^+i = 0 J V + I = cyv+i = 0. Assume that the invariant is non-zero: ( )*p =(3; 1,1 — a2,..., aw-i — ajv, fljv, 1,1 - h, • • • ,bN-i - &/v,&JV, 1,1 - C 2 , . . • , C A T _ l - C N , C N ) * Then, by Lemma 13, the coefficient of each ei,fi,gi is non-negative. Thus 1 > a,2 > • • • > ClN > 0, 1 > b2 > ••• > bN > 0, 1 > c 2 > • • • > C A T > 0. Therefore we compute 1,0,...,0,1, 1,0,. ..,0,1)* = ( 3 ; l , l , l , l , l , l ) f , where again the last equality follows from Corollary 17. 5.3 Proof of Theorem 3 Assume that the invariant is non-zero: . ^ b ; c ( C ) = ( )lP = (ai;ai, a\ - a 2,...,ajv-i - a/v, «N) ^0 Here 1" continues to denote the space constructed in section 3.3. Y 43 By Corollary 14 the multiplicities are decreasing. o-i > 0-2 > • • • > a-N > 0 Therefore, as ci\ > 0, there exists some 1 < j < N such that ax > «2 > • • • > cij > 0 cij+i = • • • = a/v = 0 Then by Corollary 17 we compute Ni,b;c(C) = (ai;ai,ai - a 2 , . . . , a j _ i - a,-, a,, 0,..., 0) y j - - i = (ai;ai,ai - a2,..., a^-i -aj,cij) Note that, for any 1 < i < j + 1, we may reorder / \Yj+1 (ar, ax, ai - a2,...., aj-i - aj, a,j)g = (ai; cti, a,i — aj+i, 0,0, a\ — a2,..., a j _ 2 — — at+2, • • •, 0,-1 — aj,aj) Applying Cremona invariance (Theorem 18) we compute ( )I/3+ = ~~ 2(°i _ «i - («i - o-i+i), 0, a i + i - aj, a i + i - a*, y j + l ' ai — a2,..., aj-i — aj, aj) Then by Lemma 13, aj+i > aj. Since this holds for all 1 < i < j we have a\ < • • • < aj. Therefore a\ = • • • — aj Thus the invariant reduces to < ) ^ = < a ; M , " - , 0 , a ) f + 1 = {a;a,a)g • 44 Appendix A Calabi-Yau configurations in blowups of P 3 A . l Another geometric construction In this appendix we construct a locally Calabi-Yau configuration of rational curves C corresponding to any arbitrary finite subtree of the maximal trivalent tree. We construct these configurations as subvarieties of a space X, which is obtained via a sequence of toric blowups of P3. X — XN ™Ni XN~l 7171,-1 > ... —L> X1 —^ X° X' X" — P 3 Our rational curves will be labeled by Aa, Ba,Ca, where a is a binary number, reflecting the trivalent nature of the configuration. Curves and homology classes in intermediary spaces will have super-scripts, and their corresponding proper trans-forms in X will not. As above, the standard torus T = (C x ) 3 action on P 3 has four fixed points; we now label the four T-fixed points in X" := P 3 by p' = (1 : 0 : 0 : 0), q' = (0 : 1 : 0 : 0), r' = (0 : 0 : 1 : 0) and s' = (0 : 0 : 0 : 1). Let A", B" and C" denote the (unique, T-invariant) line in X" through the two points {p', s'}, {q', s'} and {/•', s'}, respectively. Define x'^Ux" to be the blowup of X" at the three points {p', q', r'}, and let A', B', C C X' be the proper transforms of A", B" and C. The exceptional divisor in X' over p' intersects A' in a unique fixed point; call it p. Similarly, the exceptional divisor in X' also intersects each of B' and C in unique fixed points; call them q and r. Now define y0 7TQ, y / 45 Figure A.l: The T-invariant curves in X 1 to be the blowup of X' at the three points {p, q, r}, and let A 0 , B ° , C° C X° be the proper transforms of A',B',C. Clearly X° is deformation equivalent to a blowup of P 3 at six distinct points. The T-invariant curves in X° are depicted in Fig-ure A.l, where each edge corresponds to a T-invariant curve in X°, and each vertex corresponds to a fixed point. We now construct a sequence of blowups, beginning with X°. Define po, Pi € X° to be the two fixed points of the exceptional divisor over p which are not con-tained in A 0 ; let qo,qi,ro and r\ be defined similarly, as indicated in Figure A.l. Let J 4 Q ' ^ I c denote the T-invariants curve intersecting the pairs {A°,po} and {A°,pi}, and define BQ,B®,CQ and C® analogously. Define XX^X° to be the blowup of X° at the six points {po,pi,qo,qi,ro,ri}. Let AL,BL,CL C X1 be the proper transforms of A°,B°, and C°. We iterate this process to construct X, as follows. Fix a non-negative integer TV. For any binary number a, let \a\ denote its length, i.e. if a = a\ • • • a\ where ai G {0,1}, then |a| = I. Define to be the blowup of X 7 - 1 along the 3 x 2J points {pa, qa, ra}\a\=y Let A 3 A , BJA, CA C X-7 denote the proper transforms of A^^B-JF1 and CA~ for all \a\ < j. Let Pad and pai denote the two fixed points of the exceptional divisor ( 7 T j) - 1(p a) not intersecting A J A , and define qpo, qpi, r 7n, r 7 i similarly. Now, for each \a\ = j — 1, define Aao, A A \ to be the unique, T-invariant lines intersecting {Aa,pao},{AA^PDL}, and define Bao, BA\, Cao,cal similarly. The T-invariant curves of X 1 are shown 46 Figure A.2: The T-invariant curves in X1 in Figure A.2. Terminate this process after obtaining the space X , and define X = XN. Finally, define the configuration C c l b y C= \J Aal)BaUCa, \a\<N where A - AN n — r<N for all |a|, \B\, \^\ < N + 1. The configuration C is shown in Figure A.3, along with all other T-invariant curves in X. A . 2 Intersection products and normal bundles A.2.1 The Calabi-Yau condition We now describe H*(X,Z), identify the class of the configuration [C] & H2(X,Z), and show that C is a locally Calabi-Yau configuration in X. All (co) homology is taken with integer coefficients unless otherwise noted. Let H" denote the class of a hyperplane in X" = P3, and let H e H^(X) denote the class of its proper 47 /f' h k V Figure A.3: The T-invariant curves in X 48 transform. We may slightly abuse notation by not distinguishing between a sub-scheme and its homology class. Let E',F',G' G H 2 P O denote the exceptional divisors in X' —> X" over the points p',q' and r', and let E',F',G' G H±(X) de-note their proper transforms. In addition, let E^, Faa\ G$ G ii/"4(Xl al) denote the classes of the exceptional divisors X^ —> X l a ' _ 1 over the points pa,Qa,ra, and let Ea,Fa,Ga G H$(X) denote their proper transforms. The collection of all such classes {H, E', F', G', Ea, Fp, G 7} span HA{X). Similarly, let h" G H2(X") denote the class of a line in H", and let h G i?2p0 denote its proper transform. Also, let e',f',g' denote the class of a line in E',F',G', and let e',f',g' denote the class of their proper transforms in X. Similarly, let elal, ft[,9lal G H4{X^) denote the class of a line in E^,Faa[,G]S\ and denote their proper transforms by ea, fa, ga G H^(X). Then iJ 2(X) is generated by {Ke1 J^g1 ,ea,fp,g-,}. The intersection product ring structure is given as follows. Note that X is deformation equivalent to the blowup of P 3 at 3 + YJJ=O 3 X 2-? = 3 X 2n+1 distinct points. Therefore, these H •H = h H •h = pt E' E' = -e' E' • e = -pt F' F' = -f F' r = -pt a G' = -9' G' 9' = -pt Ea • Ea = Ea • &a = -pt Fa- Fa = -fa Fa- fa. = -pt Ga • ga = -ga Ga • 9a = -pt are all of the nonzero intersection products in H*(X). any a = ai-- •011, h-e' -- e if I = 0 &a.\•••a/_ 1 ~~ ea if 0 < I < N h-f -/ if I = 0 fa\--ai_ 1 — fa if 0 < I < N h-g'- -9 if I = 0 9a\-ai- 1 ~9a if 0 < I < N 49 Thus, we compute Kx • [C] = UH- 2{E' + F' + G')-2 Y-Ea + Fa + Ga \ \a\<N {{h-e'-e) + {h-f-f) + (h-g'-g)+ 'z ], (fiai—ai-i — 6 a + fai-oti_i fa "f~ 9ot\---ai—i 9a) J 0<KJV / = (4 - 2 - 2) + (4 - 2 - 2) + (4 - 2 - 2)+ -2 ((1-1) + (1-1) + (1-1)) 0<1<N = 0 Therefore C is a locally Calabi-Yau configuration in X. A . 2 . 2 N o r m a l bundles We now describe the normal bundles of the components of C in X. These are given as follows. NAa/X = NBa/X = NCa/X O P I(-1) ©Opi(-l) if|a| = 0 OPI © C P i ( - 2 ) ifO<|a|<iV. The equivalence NAa/X = NBa/x = ^ca/x is easily seen by relabeling points. To calculate NA/x, let D'[ c X" be a plane containing the line A", and let D\ denote its proper transform in X. Then A c D\, and NA/Dl is a sub bundle of NA/X of degree A • A. Note that D\ is deformation equivalent to the blowup of a plane at two points, and [A] — h — e' — e. Thus, the intersection product in F)\ is given by A • A = (h - e' - e) • (h - e' - e) = -1. The set of planes D\ containing A span NA/X, and the above argument holds for any such plane, so we conclude NA/X — 0(—1) © 0(—1). Now consider Aa, where a = a.\ • • • a.\ and I > 1. Note that Aa C E0il...0ll_1. As above, NAa/Eai...ai t *s a S U D bundle of NAajx of degree j4 a • A a, where the product is taken in E^...^^. Recall that, by the functoriality of blowups, Ea is the blowup of'P2 at a point, and that eai...ai_1 is the class of a line in Eai...ai_1, and ea is the exceptional divisor. We compute Aa • Aa — (eai--ai—i &a) ' ifiai-ai—i &a) = 1 1 — 0. 50 We now show that the total degree of the normal bundle is -2, forcing the result to hold. Inspect the defining exact sequence 0 - TAa -+ Tx - NAa/x - 0. This implies ci (NAa/x)=cl (Tx)-[Aa}-Cl (TAa) i AH- 2{E' + F' + G')-2 ]T Ea + Fa + G0 \a\<N ( c a i - a ( _ l — 6 Q ) 2pt = 0 - 2{pt - pt) - 2pt Thus the total degree of the normal bundle is -2, and so NAa/x = 0(a) © 0(b), where a + b = —2. Since we have already shown that (without loss of generality) a = 0, we conclude NAi/x^O®0{-2). 51 Bibliography [1] Mina Aganagic, Albrecht Klemm, Marcos Marino, and Cumrun Vafa. The topological vertex. Preprint, hep-th/0305132. [2] K. Behrend. Gromov-Witten invariants in algebraic. geometry. Invent. Math., 127(3):601-617, 1997. [3] K. Behrend and B. Fantechi. The intrinsic normal cone. Invent. Math., 128(l):45-88, 1997. [4] Jim Bryan. Multiple cover formulas for Gromov-Witten invariants and BPS states. 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[21] Jun Li, Melissa Chiu-chu Liu, Kefeng Liu, and Jian Zhou. A mathematical theory of the'closed topological vertex. Preprint, AG/0408426. [22] Jun Li and Gang Tian. Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties. J. Amer. Math. Soc, 11(1):119-174, 1998. 53 [23] R. Pandharipande. Hodge integrals and degenerate contributions. Comm. Math. Phys., 208(2):489-506, 1999. [24] Edward Witten. Two-dimensional gravity and intersection theory on moduli space. In Surveys in differential geometry (Cambridge, MA, 1990), pages 243-310. Lehigh Univ., Bethlehem, PA, 1991. 54 

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