The local Gromov–Witten invariants of configurations of rational curves
DAGANKARP
CHIU-CHUMELISSALIU
MARCOSMARIÑO
We compute the local Gromov–Witten invariants of certain configurations of rational curves in a Calabi–Yau threefold. These configurations are connected subcurves of the “minimal trivalent configuration”, which is a particular tree ofP1’s with specified formal neighborhood. We show that these local invariants are equal to certain global or ordinary Gromov–Witten invariants of a blowup of P3 at points, and we compute these ordinary invariants using the geometry of the Cremona transform. We also realize the configurations in question as formal toric schemes and compute their formal Gromov–Witten invariants using the mathematical and physical theories of the topological vertex. In particular, we provide further evidence equating the vertex amplitudes derived from physical and mathematical theories of the topological vertex.
14N35; 53D45
1 Introduction
Let Z be a closed subvariety of a smooth projective threefold X such that X is a local Calabi–Yau threefold near Z. In some cases, the contribution to the Gromov–
Witten invariants ofX by maps to Z can be isolated and defines local Gromov–Witten invariants ofZ in X. Information obtained from the study of local Gromov–Witten theory can be used to gain insight into Gromov–Witten theory in general. This has led to a great amount of interest in the subject.
The study of the local invariants of curves in a Calabi–Yau threefold has a particularly rich history. Their study goes back to the famous Aspinwall–Morrison formula for the local invariants of a single P1 smoothly embedded in a Calabi–Yau threefold with normal bundle O. 1/˚O. 1/; this result is studied by Aspinwall and Morrison [3], Cox and Katz [9], Faber and Pandharipande [10], Kontsevich [18], Lian, Liu and Yau[23], Manin[27], Pandharipande[31], and Voisin[32]. The local invariants of nonsingular curves of any genus have been completely determined by Bryan and Pandharipande[6;7;8]. In[5], Bryan, Katz and Leung computed local invariants of
certain rational curves with nodal singularities, and in particular, contractible ADE configurations of rational curves. The local invariants of the closed topological vertex, which is a configuration of three P1’s meeting in a single triple point, were computed by Bryan and the first author[4].
In this paper, we will compute local invariants of certain configurations of rational curves. The configurations considered in this paper are all connected subtrees of theminimal trivalent configuration, which is a configuration of three chains of P1’s meeting in a triple point (see Figure 1 below). A precise description of the formal neighborhood will be given inSection 3.
A1
A2
B1
B2
C1
C2
Figure 1: The minimal trivalent configuration YN DSN
iD1Ai[Bi[Ci. The normal bundles ofA1;B1;C1are isomorphic toO. 1/˚O. 1/; the normal bundle of any other irreducible component is isomorphic to O˚ O. 2/.
1.1 Local Gromov–Witten invariants
Let ZX be a closed subvariety of a smooth projective Calabi–Yau threefold. Let Mg.X;d/ denote the stack of genusg stable maps toX representing d2H2.X;Z/.
It is a Deligne–Mumford stack with a perfect obstruction theory of virtual dimension zero which defines a virtual fundamental zero-cycle ŒMg.X;d/vir.
Whenever the substack Mg.Z/consisting of stable maps whose image lies in Z is a union of path connected components of Mg.X;d/, it inherits a degree-zero virtual class. The genus-g local Gromov–Witten invariant of Z in X is defined to be the degree of this virtual class, and is denoted by Ndg.ZX/. We writeNdg.Z/ when the formal neighborhood is understood.
We will consider genus g, degree dlocal Gromov–Witten invariants Ndg.YN/, where
dD
N
X
jD1
d1;jŒAjCd2;jŒBjCd3;jŒCj
2H2.YNIZ/:
For simplicity, we write dD.d1;d2;d3/where diD.di;1; : : : ;di;N/. In this paper, we always assume d iseffectivein the sense thatdi;j 0. We will show that the local invariants Ndg.YN/ are well defined in the following cases:
(i) (The minimal trivalent configuration) d1;1Dd2;1Dd3;1D1.
(ii) (A chain of rational curves) d1;1>0, d2;j Dd3;j D0 for 1j N.
We will see inSection 3that the formal neighborhood of YN has a cyclic symmetry, so one can cyclically permute d1;d2;d3 in Case (ii). We show that in the above cases the local invariants Ndg.YN/ are equal to certain global or ordinary Gromov–Witten invariants of a blowup of P3 at points (Section 4), and we compute these ordinary invariants using the geometry of the Cremona transform (Section 2). To state our results, define constants Cg by
(1)
1
X
gD0
Cgt2gD
t=2 sin.t=2/
2
D
1
X
gD0
jB2g.2g 1/j .2g/! t2g:
Theorem 1 (The minimal trivalent configuration) Suppose that d1;1Dd2;1Dd3;1D1:
Then
Ndg.YN/D
Cg if1Ddi;1 di;N 0for i D1;2;3; 0 otherwise.
Theorem 2 (A chain of rational curves) Suppose that
d1D.d1; : : : ;dN/; d2Dd3D.0; : : : ;0/;
whered1>0. Then
Ndg.YN/D 8 ˆ<
ˆ:
Cgd2g 3 ifd1Dd2D Ddk Dd>0and
dkC1DdkC2D DdN D0for some1kN
0 otherwise.
Our results are new and add to the list of configurations of rational curves for which the local Gromov–Witten invariants are known.
The configuration inTheorem 2is an AN curve. It is interesting to compareTheorem 2 with the result for a generic contractible AN curve E D E1[ [EN from Bryan–Katz–Leung[5, Proposition 2.10]:
Fact 1 (A generic contractibleAN curve[5]) Assumedi>0 foriD1; : : : ;N. Let Ng.d1; : : : ;dN/ denote genus g local Gromov–Witten invariants of E in the class PN
jD1djŒEj. Then
Ng.d1; : : : ;dN/D
Cgd2g 3 d1D DdN Dd>0;
0 otherwise.
Note that Y1 is the closed topological vertex. By the results in Faber–Pandharipande [10]and Bryan–Karp[4], Ndg
1;d2;d3.Y1/ is defined in the following cases:
(iii) (Super-rigid P1) d1>0;d2Dd3D0 (and its cyclic permutation).
(iv) (The closed topological vertex) d1;d2;d3>0. Fact 2 (Super-rigidP1 [10]) Suppose that d>0. Then
Ndg;0;0.Y1/DN0g;d;0.Y1/DN0g;0;d.Y1/DCgd2g 3:
Fact 3 (The closed topological vertex[4]) Suppose thatd1;d2;d3>0. Then Ndg
1;d2;d3.Y1/D
Cgd2g 3 d1Dd2Dd3Dd>0;
0 otherwise.
1.2 Formal Gromov–Witten invariants
The minimal trivalent configurationYN together with its formal neighborhood is a nonsingular formal toric Calabi–Yau (FTCY) scheme YyN. Theformal Gromov–Witten invariants Nzdg.YyN/ ofYyN are defined for all nonzero effective classes (seeSection 5.1and Bryan–Pandharipande[8, Section 2.1]). Moreover,
Nzdg.YyN/DNdg.YN/
in all the above cases (i)–(iv). Introduce formal variables ;ti;j and define ZzN.It/Dexp X
g0
X
d
2g 2Nzg;d.YyN/e dt
where d runs over all nonzero effective classes, and tD.t1;t2;t3/; ti D.ti;1; : : : ;ti;N/; dtD
3
X
iD1 N
X
jD1
di;jti;j:
We call ZzN.It/ thepartition functionof formal Gromov–Witten invariants of YyN. It is the generating function ofdisconnectedformal Gromov–Witten invariants ofYyN.
InSection 5, we will compute ZzN.It/ by the mathematical theory of the topological vertex (see Li–Liu–Liu–Zhou[22]) and get the following expression (Proposition 17):
(2) ZzN.It/Dexp 1
X
nD1
1 nŒn2
3
X
iD1
X
2k1k2N
e n.ti;k1CCti;k2/
X
E
WzE.q/
3
Y
iD1
. 1/jije jijti;1s.i/t.ui.q;ti//:
where E D.1; 2; 3/is a triple of partitions, qDe
p 1,ŒnDqn=2 q n=2. The precise definitions of WzE.q/ and s.i/t.ui.q;ti// will be given inSection 1.3. In particular, we will show that
(3) Zz1.It/DX
E
WzE.q/
3
Y
iD1
. 1/jije jijtiW.i/t.q/Dexp
1
X
nD1
Qn.t/
nŒn2
!
where tD.t1;t2;t3/, W.q/ is defined by(9)inSection 1.3, and (4) Qn.t/D
e nt1Ce nt2Ce nt3 e n.t1Ct2/ e n.t2Ct3/ e n.t3Ct1/Ce n.t1Ct2Ct3/: InSection 6, we will computeZzN.It/by the physical theory of the topological vertex (see Aganagic–Klemm–Marino–Vafa˜ [2]) and get the following expression (Proposition 20):
(5) ZN.It/Dexp 1
X
nD1
1 nŒn2
3
X
iD1
X
2k1k2N
e n.ti;k1CCti;k2/
X
E
WE.q/
3
Y
iD1
. 1/jije jijti;1s.i/t.ui.q;ti//
where WE.q/ is defined by(8)inSection 1.3. In particular, we will show that (Propo- sition 19):
(6) Z1.It/DX
E
WE.q/
3
Y
iD1
. 1/jije jijtiW.i/t.q/Dexp
1
X
nD1
Qn.t/ nŒn2
!
The equivalence of the physical and mathematical theories of the topological vertex boils down to the following combinatorial identity:
(7) W1;2;3.q/D zW1;2;3.q/:
It is known that(7)holds when one of the three partitions is empty (see the work of Li, C-C M Liu, K Liu and Zhou[24;22]). When none of the partitions is empty, Klemm has checked all the cases where jij 6 by computer. Up to now, a mathematical proof of(7)in full generality is not available. Equations(3)and(6)imply the following result.
Theorem 3 X
E
WE.q/
3
Y
iD1
. 1/jije jijtiW.i/t.q/DX
E
WzE.q/
3
Y
iD1
. 1/jije jijtiW.i/t.q/:
Theorem 3provides further evidence of (7)equating the vertex amplitudes derived from physical and mathematical theories of the topological vertex.
1.3 The topological vertex
In[2], Aganagic, Klemm, Marino, and Vafa proposed that Gromov–Witten invariants of˜ any toric Calabi–Yau threefold can be expressed in terms of certain relative invariants of its C3 charts, calledthe topological vertex. They suggested that these local relative invariants should count holomorphic maps from bordered Riemann surfaces toC3where the boundary circles are mapped to three explicitly specified Lagrangian submanifolds L1;L2;L3. The topological vertex depends on three partitions E D.1; 2; 3/, where i corresponds to the winding numbers (the homology classes of boundary circles) inLi. There is a symmetry on C3 cyclically permutingL1;L2;L3, so one expects the topological vertex to be symmetric under a cyclic permutation of the three partitions 1; 2; 3.
In[2]the topological vertex was computed by using the conjectural relation between open Gromov–Witten invariants on toric Calabi–Yau threefolds and Chern–Simons invariants of knots and links. It has the following form:
(8) WE.q/Dq2=2C3=2 X
;1;3
c1
1c.3/t
3
W.2/t1.q/W23.q/ W2.q/ :
In(8), t denotes the partition transposed to . The expression(8)involves various quantities that we now define. is given by
DX
i
i.i 2iC1/:
The coefficientsc are Littlewood–Richardson coefficients. They can be defined in terms of Schur functions as follows
ssDX
c s:
Here, Schur functions are regarded as a basis for the ring ƒof symmetric polynomials in an infinite number of variables. The quantityW.q/ can be also defined in terms of Schur functions as follows:
(9) W.q/Ds xiDq iC12
: One can show that
(10) Wt.q/Dq =2W.q/:
We also define, in analogy to skew Schur functions,
(11) W=.q/DX
c W.q/:
Finally, W.q/ is defined by
(12) W.q/Dq=2C=2X
Wt=.q/Wt=.q/:
This expression for W.q/ is different from the one used originally in[2]. The fact that both agree follows from cyclicity of the vertex, and it has been proved in detail by Zhou[33]. The expression for the vertex in terms of Schur functions is given in Okounkov–Reshetikhin–Vafa[30]where the cyclicity of the vertex is also proved.
In Li–Liu–Liu–Zhou[22]the topological vertex was interpreted and defined as local relative invariants of a configurationC1[C2[C3of threeP1’s meeting at a pointp0in a relative Calabi–Yau threefold.Z;D1;D2;D3/, whereKZCD1CD2CD3ŠOZ, Ci intersectsDi at a pointpi¤p0, andCi\Dj is empty fori¤j. The partitioni corresponds to the ramification pattern overpi. It is shown in[22]that Gromov–Witten invariants of any toric Calabi–Yau threefold (or more generally, formal Gromov–Witten invariants of formal toric Calabi–Yau threefolds) can be expressed in terms of local relative invariants as described above, and the gluing rules coincide with those stated
in Aganagic–Klemm–Mari˜no–Vafa[2]. The following expression of the vertex was derived in[22]:
(13) WzE.q/Dq .1 22 123/=2 X
C;1;3;1;3
c.C1/t2c1
.1/t1c3
3.3/t
q. 2C
3
2 /=2WC3.q/X
1
z1./3.2/:
Here2D.2122 / if D.12 /. Recall that z DY
i1
imimi!
where miDmi./is the number of parts of the partition equal to i (see Macdonald [26, p.17]).
It is expected that the two different enumerative interpretations in[2]and in[22]of the vertex give rise to equivalent counting problems, in the spirit of the following simple example: counting ramified covers of a disc by bordered Riemann surfaces with prescribed winding numbers is equivalent to counting ramified covers of a sphere by closed Riemann surfaces with prescribed ramification pattern over 1.
Finally, we introduce some notation which will arise in computations inSection 6. For any positive integern, define
(14) uin.q;ti/D 1 Œn
1C
N
X
kD2
e n.ti;2CCti;k/ :
Given a partition D.12 `>0/, define
(15) ui.q;ti/D
Y` jD1
ui
j.q;ti/:
and
(16) s.ui.q;ti//D X
jjDjj
./
z ui.q;ti/:
In particular, when N D1, we have uinD 1
Œn DX
i>0
q iC1=2
So
(17) s.ui.q;ti//Ds.xiDq iC12/DW.q/:
Acknowledgements
The authors give warm thanks to Jim Bryan for helpful conversations. The first author thanks NSERC for its support. The second author thanks Jun Li, Kefeng Liu, Jian Zhou for collaboration[22]and Shing-Tung Yau for encouragement. Finally, the authors thank the referee for the detailed comments and numerous valuable suggestions on the presentation.
2 Cremona
In this section we prove Theorems1and2using the geometry of the Cremona transform.
We assume that the formal neighborhood YN X is as constructed inSection 3. We also assume that the local invariants ofYN are equal to certain ordinary invariants of X, which we prove inSection 4.
2.1 The blowup of CP3 at points
We briefly review the properties of the blowup of P3 at points used here for complete- ness and to set notation. This material can be found in much greater detail in, for instance, Griffiths–Harris[13].
LetX!P3 be the blowup ofP3 alongM distinct pointsfp1; : : : ;pMg. 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 Poincar´e 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 exceptional divisor over pi. Then H4.X;Z/ has a basis
H4.X/D hH;E1; : : : ;EMi:
Furthermore, let h2H2.X/ be the class of a line inH, and let ei be the class of a line inEi. The collection of all such classes form a basis of H2.X/.
H2.X/D hh;e1; : : : ;eMi
The intersection ring structure is given as follows. Letpt2H0.X/ denote the class of a point. Two general hyperplanes meet in a line, so HHDh. A general hyperplane
and line intersect in a point, soHhDpt. Also, a general hyperplane is far from the center of a blowup, so all other products involving H orh vanish. The restriction of OX.Ei/ to Ei ŠP2 is the dual of the bundle OP2.1/, so EiEi is represented by minus a hyperplane inEi, i.e. EiEiD ei, andEi3D. 1/3 1ptDpt (see Fulton [11]). 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.
HHDh HhDpt EiEiD ei EieiD pt
Also, we point out the that the canonical bundle KX is easy to describe in this basis:
KX D 4HC2
M
X
iD1
Ei
Finally, we introduce a notational convenience for the Gromov–Witten invariants of P3 blown up at points in a Calabi–Yau class. Any curve class is of the form
ˇDdh
M
X
iD1
aiei
for some integers d;ai where d is non-negative. Thus KX ˇ D0 if and only if 2dDPM
iD1ai. In that case, the virtual dimension of Mg.X; ˇ/ is zero, and h iXg;ˇD
Z
ŒMg.X;ˇ/vir1
is determined by the discrete data fd;ai; : : : ;aMg. Then, we may use the shorthand notation
h iXg;ˇD hdIa1; : : : ;aMiXg : For example,
h iXg;5h e1 e2 2e3 3e5 3e6D h5I1;1;2;0;3;3iXg :
Furthermore, the Gromov–Witten invariants of X do not depend on ordering of the points pi, and thus for any permutation ofM points,
hdIa1; : : : ;aMiXg D˝
dIa.1/; : : : ;a.M/˛X g :
2.2 Properties of the invariants of the blowup ofP3 at points
First, we use the fact, shown in Bryan–Karp[4], that the Gromov–Witten invariants of the blowup of P3 along points have a symmetry which arises from the geometry of the Cremona transformation.
Theorem 4 (Bryan–Karp[4]) Let ˇDdh PM
iD1aiei with 2d DPM
iD1ai and assume thatai¤0for somei >4. Then we have the following equality of Gromov–
Witten invariants:
h iXg;ˇD h iXg;ˇ0
whereˇ0Dd0h PM
iD1a0iei has coefficients given by d0 D3d 2.a1Ca2Ca3Ca4/ a01D d .a2Ca3Ca4/ a02D d .a1Ca3Ca4/ a03D d .a1Ca2Ca4/ a04D d .a1Ca2Ca3/ a05D a5
:::
aM0 D aM:
We also use the following vanishing lemma, and a few of its corollaries.
Lemma 5 Let X be the blowup ofP3 at M distinct generic points fx1; : : : ;xMg, and ˇDdh PM
iD1aiei with2dDPM
iD1ai, and assume thatd>0 andai<0 for somei. Then
Mg.X; ˇ/D∅:
Corollary 6 For anyM points fx1; : : : ;xMgandX and ˇ as above the correspond- ing invariant vanishes;
h iXg;ˇD0:
This follows immediately from the deformation invariance of Gromov–Witten invariants andLemma 5.
Proof In genus zero,Lemma 5follows from a vanishing theorem of Gathmann[12, Section 3]. In order to proveLemma 5, for arbitrary genus, it suffices to show that the result holds for a specific choice of points, as if the moduli space is empty for a specific choice, then it is empty for the generic choice. By choosing some of the points to be coplanar, and the rest to also be coplanar on a second plane, the result follows.
For further details, see Karp[17].
Corollary 7 Let X be the blowup of P3 along M points and define ˇ Ddh PM
iD1aiei where2dDPM
iD1ai andd >0. Also define X0! X
to be the blowup ofX at a generic pointp, so that X0is deformation equivalent to the blowup of P3 atM C1distinct points. Letfh0;e10; : : : ;eM0 C1gbe a basis of H2.X0/, and letˇ0Ddh0 PM
iD1aiei0. Then
hdIa1; : : :aM;0iXg0D hdIa1; : : : ;aMiXg
Proof This result follows from the more general results of Hu[14]. An independent proof usingLemma 5can be found in Karp[17].
2.3 Proof ofTheorem 1
Let the blowup space XNC1 and the minimal trivalent configurationYN be as con- structed inSection 3on page128. ByProposition 8on page133we have
Ndg.YN/D h iXg;NdC1: Assume that the invariant is non-zero:
h iXg;dNC1D˝
3I1;1 d1;2; : : : ;d1;N 1 d1;N;d1;N; 1;1 d2;2; : : : ;d2;N 1 d2;N;d2;N; 1;1 d3;2; : : : ;d3;N 1 d3;N;d3;N˛XNC1
g
¤0
Then, byCorollary 6 , the coefficient of each ei; fi;gi is non-negative. Thus, for iD1;2;3,
(18) 1di;2 di;N 0:
Therefore we compute
h iXg;dNC1D˝
3I1;0; : : : ;0;1; 1;0; : : : ;0;1; 1;0; : : : ;0;1˛XNC1
g
D h3I1;1;1;1;1;1iXg2;
where the last equality follows fromCorollary 7. So when(18)holds, we have Ndg.YN/DN1g;1;1.Y1/DCg:
The last equality follows fromFact 3(see Bryan–Karp[4]).
2.4 Proof ofTheorem 2
Let the blowup space XzNC1 and the chain of rational curvesYAN be as constructed in Section 3on page130. ByProposition 10on page137we have
Ndg.YN/DNdg
1.YAN/D h iXgz;dNC1
where
d1D.d1; : : : ;dN/; d2Dd3D.0; : : : ;0/:
Assume that the invariant is non-zero:
h iXgz;NdC1D hd1Id1;d1 d2; : : : ;dN 1 dN;dNiXgzNC1¤0: ByCorollary 6the multiplicities are decreasing:
d1d2 dN 0 Therefore, asd1>0, there exists some1j N such that
d1 dj >0; djC1D DdN D0: Then, usingCorollary 7, we compute
Ndg.YN/D˝
d1Id1;d1 d2; : : : ;dj 1 dj;0; : : :0˛XzNC1 g
D˝
d1Id1;d1 d2; : : : ;dj 1 dj;dj
˛XzjC1
g :
Note that for any 1ijC1 we may reorder
˝d1Id1;d1 d2; : : : ;dj 1 dj;dj˛XzjC1
g D
hd1Id1;d1;di diC1;0;0;d1 d2; : : :
: : : ;di 2 di 1;diC1 diC2; : : : ;dj 1 dj;dj˛XzjC1 g
Applying Cremona invariance (Theorem 4) we compute h iXgz;dNC1D˝
d1 2.di diC1/Id1 .di diC1/;0;diC1 di;diC1 di;
d1 d2; : : : ;dj 1 dj;dj˛XzjC3
g :
Then, byCorollary 6, diC1di. Since this inequality holds for every 1i j we have d1 dj. Therefore
d1D Ddj Dd: Thus we have
h iXgz;NdC1D hdId;0; : : : ;0;diXgzjC1
D hdId;diXgz2
DNdg;0;0.Y1/ DCgd2g 3
The last equality follows from Faber–Pandharipande[10].
3 Construction
We construct these configurations as subvarieties of a locally Calabi–Yau spaceXNC1, which is obtained via a sequence of toric blowups of P3:
XNC1 NC!1 XN N! !2 X1 !1 X0DP3
In fact, XiC1 will be the blowup ofXi along three points. Our rational curves will be labeled byAi;Bi;Ci, where 1iN, reflecting the nature of the configuration.
Curves in intermediary spaces will have super-scripts, and their corresponding proper transforms in X will not.
The standard torusTD.C/3 action onP3 is given by
.t1;t2;t3/.x0Wx1Wx2Wx3/7!.x0Wt1x1Wt2x2Wt3x3/:
There are four T–fixed points in X0W DP3; we label them p0D.1W0W0W0/, q0D .0W1W0W0/, r0 D.0W0W1W0/ and s0 D.0W0W0W1/. Let A0, B0 and C0 denote the (unique,T–invariant) line inX0through the two points fp0;s0g,fq0;s0gandfr0;s0g, respectively.
Define
X1 !1 X0
to be the blowup of X0 at the three points fp0;q0;r0g, and let A1;B1;C1 X1 be the proper transforms of A0;B0 andC0. The exceptional divisor inX1 over p0
intersectsA1 in a unique fixed point; call itp12X1. Similarly, the exceptional divisor inX1 also intersects each of B1 and C1 in unique fixed points; call themq1 and r1.
A21 A22
p2 B21
B22 q2
C21
C22 r2
Figure 2: TheT–invariant curves inX2
Now define
X2 !2 X1
to be the blowup of X1 at the three points fp1;q1;r1g, and let A21;B12;C12X2 be the proper transforms ofA1;B1;C1. The exceptional divisor over p1 contains two T–fixed points disjoint fromA21. Choose one of them, and call it p2; this choice is arbitrary. Similarly, there are two fixed points in the exceptional divisors aboveq1;r1 disjoint from B12;C12. Choose one in each pair identical to the choice of p2 and call them q2 and r2 (identical makes sense here as the configuration of curves inFigure 2is rotationally symmetric). This choice is indicated inFigure 2. LetA22 denote the (unique, T–invariant) line intersectingA21 andp2. Define B22;C22 analogously.
Clearly X2 is deformation equivalent to a blowup of P3 at six distinct points. The T–invariant curves in X2 are depicted inFigure 2, where each edge corresponds to a T–invariant curve in X2, and each vertex corresponds to a fixed point.
A31 A32
A33
p3 B13
B32 B3
3 q3
C31 C23 C33 r3
Figure 3: TheT–invariant curves inX3
We now define a sequence of blowups beginning with X2. Fix an integer N 2. For each 1<i N, define
XiC1 iC!1 Xi
to be the blowup ofXi along the three pointspi;qi;ri. Let AjiC1XiC1 denote the proper transform ofAji for each1ji. The exceptional divisor inXiC1 abovepi
contains twoT–fixed points, choose one of them and call itpiC1. Similarly choose qiC1;riC1, and define AiiCC11XiC1 to be the line intersectingAiiC1 and piC1, with BiiCC11;CiiCC11 defined similarly. The T–invariant curves in X3 are shown inFigure 3.
Finally, we define theminimal trivalent configuration YN XNC1 by YN D [
1jN
Aj[Bj[Cj; where
Aj DAjNC1; Bj DBjNC1; Cj DCjNC1:
The configurationYN is shown inFigure 4, along with all otherT–invariant curves in XNC1. It contains a chain of rational curves:
YAN DA1[ [AN:
ŒA1Dh e1 e2
ŒA2De2 e3
ŒA3De3 e4
ŒA4De4 e5
ŒANDeN eNC1
eNC1
e2
e3
e4
eNC1
e2 e3 e3 e4 e5 e6 eN eNC1 eNC1
e1 e2
e1
e1 eNC1
ŒB1Dh f1 f2
ŒB2Df2 f3
ŒB3Df3 f4
ŒB4Df4 f5
ŒBNDfN fNC1
fNC1
f2
f3
f4
fNC1
f2 f3
f3 f4
f5 f6
fN fNC1
fNC1
f1 f2
f1
f1 fNC1
ŒC1Dh g1 g2
ŒC2 Dg2
g3
ŒC3 Dg3
g4
ŒC4 Dg4
g5
ŒCN DgN
gNC1
gNC1
g2
g3
g4
gNC1
g2 g3
g3 g4
g5 g6
gN gNC1
gNC1
g1 g2
g1
g1
gNC1
h e1 f1
h e1 g1
h f1 g1
Figure 4: TheT–invariant curves inXNC1
3.1 Homology
We now compute H.XNC1;Z/ and identify the class of the configurationŒYN2 H2.XNC1;Z/. All (co)homology will be taken with integer coefficients. We denote divisors by 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 inXN are undecorated.
LetEz1;Fz1;Gz12H4.X1/denote the exceptional divisors inX1!X0over the points p0;q0 and r0, and let E1;F1;G12H4.X/denote their total transforms. Continuing,
for each 1iNC1, let Ezi;Fzi;Gzi2H4.Xi/ denote the exceptional divisors over the points pi 1;qi 1;ri 1 and let Ei;Fi;Gi2H4.X/ denote their total transforms.
Finally, letH denote the total transform of the hyperplane inX0DP3. The collection of all such classes fH;Ei;Fi;Gig, where1iNC1, spans H4.XNC1/. Similarly, for each 1iNC1, let zei;fzi;gzi2H2.XiC1/denote the class of a line inEzi;Fzi;Gzi and let ei; fi;gi2H2.X/denote their total transforms. In addition, let h2H2.XNC1/ denote the class of a line in H. ThenH2.XNC1/ has a basis given by fh;ei; fi;gig.
The intersection product ring structure is given as follows. Note thatXNC1 is defor- mation equivalent to the blowup of P3 at3N distinct points. Therefore, these
HHDh HhDpt EiEiD ei EieiD pt
FiFiD fi FifiD pt GiGiD gi GigiD pt are all of the nonzero intersection products inH.XNC1/.
In this basis, the classes of the components ofYN are given as follows.
ŒAiD
(h e1 e2 ifiD1 ei eiC1 otherwise ŒBiD
(h f1 f2 ifiD1 fi fiC1 otherwise ŒCiD
(h g1 g2 ifiD1 gi giC1 otherwise
To see this, recall thatA1 is the proper transform of a line through two points which are centers of a blowup, and that Ai, fori>1, is the proper transform of a line in an exceptional divisor containing a center of a blowup. Bi andCi are similar.
4 Local to global
In this section, we will show that the local invariants Ndg.Y/ are equal to the ordinary invariants h iXg;NdC1 in case Y is either the minimal trivalent configuration YN or the chain of rational curves YAN defined inSection 3.
4.1 The minimal trivalent configuration
Proposition 8 LetfW †!XNC1 represent a point inMg.XNC1;d/, where 1Ddi;1 di;N 0:
Then the image off is contained in the minimal trivalent configuration YN D [
1jN
Aj[Bj[Cj:
Proof We use the toric nature of the construction. Assume that there exists a stable map
ŒfW †!XNC12Mg.XNC1;d/
such that Im.f /6YN. Then there exists a pointp2Im.f / such thatp62YN. Recall that T–invariant subvarieties of a toric variety are given precisely by orbit closures of one-parameter subgroups ofT. So in particular the limit of p under the action of a one-parameter subgroup is aT–fixed point. Moreover, sincep62YN, there exists a one-parameter subgroup W C!T such that
tlim!0 .t/pDq where q isT–fixed andq62YN.
The limit of acting on Œf is a stable mapf0 such thatq2Im.f0/. It follows that q is in the image of all stable maps in the orbit closure ofŒf0. Thus, there must exist a stable mapŒf00W †!XNC12Mg.XNC1;d/ such thatIm.f00/ isT–invariant and Im.f00/62YN.
We show that this leads to a contradiction. Let F denote the union of the T–invariant curves in XNC1; it is shown above inFigure 4. We study the possible components of F contained in the image off00.
Note that the push forward of the class of †is given by f00Œ†D 3h e1
N 1
X
jD1
.d1;j d1;jC1/ejC1 d1;NeNC1
f1 N 1
X
jD1
.d2;j d2;jC1/fjC1 d2;NfNC1
g1
N 1
X
jD1
.d3;j d3;jC1/gjC1 d3;NgNC1: