Genus Two Stable Maps, Local Equations and Modular Resolutions

0.41 MB·English

by  Yi Hu

GENUS TWO STABLE MAPS, LOCAL EQUATIONS AND
MODULAR RESOLUTIONS
2
1 YIHUANDJUNLI
0
2

n
a
J Abstract. WedescribeacanonicalsequenceofmodularblowupsoftheArtin
1 stack of stable weighted genus-two curves that diagonalize certain canonical
1 derived objects of the original Artin stack. This provides a resolution of the
primarycomponentofthemodulispaceofgenustwostablemapstoprojective
] space,andapartialdesingularizationofthesamemodulispace. Ourapproach
G shouldextend tohighergenera.
A
.
h 1. Introduction
t
a ThemodulispacesM (Pn,d)ofstablemapsintoprojectivespacehasbeenone
m g,m
ofthe centralobjects in algebraicgeometrysince its introduction by Kontsevichin
[ early90’s ([5, 1,2]). We are interestedin these spaces because Mg,m(Pn,d) carries

1 certaintautologicalderived objects and bundle stacks whose topologicalinvariants
v whenintegratedoverthevirtualfundamentalclassofM (Pn,d)givesustheGW
g,m
7 invariants of complete intersections in the projective space.
2
Tothisend,weliketofindamodular(partial)desingularizationofmodulispace
4
sothattheseobjectsandbundle stackscanbe definedandevaluated. Oursolution
2
. is to work on certain direct images sheaves (complex), study their degeneracy loci
1
and construct modular blowing-up based on these loci to make the direct image
0
2 complex diagonalizable. Once this is achieved, the partial desingularization of the
1 moduli space M (Pn,d) follows by standard argument.
g
: In this paper, we will work out the g = 2 case in details. The results proved
v
i will allow us to prove the “hyperplance” property of the GW invariants of quintic
X
Calabi-Yau threefolds conjectured in [6]. We hope this work will also shed light to
r constructing modular blowing up of higher genus moduli space M (Pn,d).
a g
WeletMwt betheArtinstackofprestableweightedcurvesofgenus2(i.e. nodal
2
curves whose irreducible components are decorated by weights ∈Z , cf. [3]); let
≥0
(1.1) ςk :M2(Pn,d)−→Mw2t; [u,C]→(C,c1(u∗OPn(k))),
be the induced representable morphism. In this paper, we construct a sequence of
modular blowups of Mwt along smooth centers: Mwt −→Mwt. We set
2 2 2
Mk(Pn,d)=M (Pn,d)× fMwt, k ≥2.
2 2 ςk,Mw2t 2
f f
Theorem1. Let(π,f,C)betheuniversalfamilyonM (Pn,d). Supposek ≥2,then
2
the pullback to M2k(Pn,d) of derived object Rπ∗f∗OPn(k) is locally diagonalizable.
For the definfition of locally diagonalizable object, see §2.
1
2 YIHUANDJUNLI
Corollary 2. Let the situation be as in Theorem 1, and let N be any irreducible
component of Mk(Pn,d) with (πk ,fk ) the pullback to N of (π,f). Then the direct
2 N N
image sheaf (πfNk )∗(fkN)∗OPn(k) is locally free.
To obtain a partial desingularization of M (Pn,d), we need the analogue of
2
Theorem 1 for k = 1. For this we need one more round of modular-blowing ups
along smooth centers in a refinement of Mwt. We let P be the relative Picard
2 2
stack over M ; its objects are (C,L), where C are genus 2 nodal curves and L are
2
line bundles over C. We have representable morphisms
(1.2) ̺k :M2(Pn,d)−→P2; [u,C]→(C,u∗OPn(1)).
Theorem 3. We have a sequence of blow-ups P →P so that if we let
2 2
M (Pn,d)=M (Pn,de)× P ,
2 2 P2 2
thenthepullback toM2(Pnf,d)ofthederived object Rπ∗ef∗OPn(1)islocally diagonal-
izable. Further, for d>2 the stack M (Pn,d) has normal crossing singularities.
f 2
To diagonalizeRπ∗f∗OPn(k), we pferformthree sequences of blow-ups of Mw2t to
obtain the smooth stack Mwt; in the case k =1, we perform one more sequence of
2
blow-ups of P × Mwt to obtain the smooth stack P . We describe briefly the
2 Mwt 2f 2
2
blowing-up centers of these sequences of blowing-ups.
f e
Our innovative approach improves the technique of [3] and combines with the
ideaofthe derivedmodularblowups in[4]. To explainthis, weintroduce the Artin
stack Mdiv of pairs (C,D) where C is a prestable genus-2 curve and D is a simple
2
divisor supported on the smooth locus of C. The stack Mdiv comes equipped with
2
a universal curve ρ : C → Mdiv and a universal divisor D; there is a canonical
2
morphism Mdiv −→Mwt, (C,D) →(C,c (O (D))); via this morphism, we study
2 2 1 C
the parallel diagonalization problem of the derived object ρ O (D). Locally over
∗ C
a small smooth chart V, the object ρ O (D)| can be represented by a two term
∗ C V
structural homomorphism ϕ : O⊕m+1 → O⊕2 where m = degD. We provide a
V V
detail analysis of this homomorphism in terms of local modular parameters. This
analysis guides to the following blowups.
The first round blowups. Let Π = [0,d]∩Z. We introduce a (first order) depth
A
function ℓ:Mwt −→Π ; for eachpoint x∈Mwt, the invariantℓ(x) quantitatively
2 A 2
measureshowfarthe homomorphismϕ:O⊕m+1 →O⊕2 frombeingofrank1. For
V V
any k ≥1, we let
Θ1 ={x∈Mwt |ℓ(x)=k}.
k 2
We thenblow upMwt along(the propertransformsof)Θ1,Θ1,··· ,Θ1. We letN
2 1 2 d 1
be the final blowup stack of the first round.
Thesecondroundblowups. LetΠ ={(i,j)∈Z×Z)|0≤j <i≤d}.We provide
B
a total ordering of Π and list its elements by its order as Π = {α ,α ,··· ,α }
B B 0 1 r
where r = d(d+1). We then construct a (second order) depth function ℓ′ : N −→
2 1
Π ; for each point x∈ N , the invariant ℓ′(x) measures how far (the pullback of)
B 2
the homomorphismϕ:O⊕m+1 →O⊕2 frombeing ofrank 2. For any k ≥1, we let
V V
Θ2 ={x∈N |ℓ′(x)=α }.
k 2 k
MODULAR RESOLUTION OF MODULI SPACE OF STABLE MAPS 3
We then blow up N along (the proper transforms of) Θ2,Θ2,··· ,Θ2. We let N
1 1 2 r 2
be the final blowup stack of this round.
After the second round, for any point x = (C,w) ∈ Mwt, its neighborhood has
2
been modified so that the pullback of the structural homomorphism ϕ becomes
diagonalizable, possibly except (1) when the smallest genus-2 subcurve F of C,
calledthe coreof C, has rationaltails attachedat conjugate points (or Weierstrass
points) of F; (2) when the core F has exactly weight 2.
The third round blowups. For the first possibility (1), we construct a (third order)
depthfunctionℓ′′ :N −→Π whereΠ =[0.d]∩Z. Therestgoesalmostthesame
2 C C
as above. This round produces our stack Mwt. We leave the second possibility (2)
2
to the last round of blowing-ups.
f
We point out here that the Mw2t diagonalizes the derived objects Rπ∗f∗OPn(k)
for all k ≥ 2 except Rπ∗f∗OPn(f1). To resolve Rπ∗f∗OPn(1) and hence a (partial)
desingularization M (Pn,d) (d > 2), we need one more round of blowups to re-
2
solve the locus of M (Pn,d) whose general points are stable maps such that their
2
restrictions to the cores are double covers of smooth rational curves.
The fourth round blowups. The procedure of this round is analogous to the third
round (i.e., relying on a third order depth function). We let H be the closed
k
substack of P whose general points are pairs (C,L) with F the core of C such
2
that degL| = 2, h0(L| ) = 2 and the core F have k many rational tails. We
F F
then blow up the stack P × Mwt along (the proper transforms of ) H ×
2 Mwt 2 1 Mwt
2 2
Mwt,··· ,H × Mwt,··· . We denote the resulting stack by P .
2 m Mwt 2 f 2
2
fOnce the Theoremf3 is established, applying the technique ine[3], we obtain the
local structure of the total transforms M (Pn,d). We will provide the details of
2
this proof in Section 7.
f
Throughoutthepaper,weworkoveranalgebraicallyclosedfieldofcharacteristic
zero.
Acknowledgement. Whilethispaperbeingprepared,thefirstnamedauthorwas
partially supported by NSF DMS 0901136;the second named author was partially
supported by NSF DMS-1104553.
2. Locally Diagonalizable Derived Objects
We begin with introducing diagonalizable derived objects, a key notion used
throughout the article.
2.1. Diagonalizable homomorphism. Diagonalizable homomorphism is intro-
duced in [4].
Definition 2.1.1. Let E and E be two locally free sheaves of O -modules on a
1 2 U
DM stack U. We say a homomorphism ϕ : E → E is diagonalizable if there are
1 2
integers r, l and l ∈ Z , p ∈ Γ(O ) such that the ideals (p ) ⊂ (p ), and
1 2 ≥0 i U i i+1
isomorphisms Ei ∼=OU⊕r⊕OU⊕li such that
ϕ=diag[p ,··· ,p ]⊕0:O⊕r⊕O⊕l1 −→O⊕r⊕O⊕l2,
1 r U U U U
4 YIHUANDJUNLI
where diag[p ,··· ,p ] : O⊕r → O⊕r is the diagonal homomoprhism; 0 : O⊕l1 →
1 r U U U
O⊕l2 is the zero homomorphism. We say ϕ is locally diagonalizable if there is an
U
´etale cover U of U such that ϕ is diagonalizable over U .
α α
It isdirectto checkthatin caseϕ:E −→F is locallydiagonalizablehomomor-
phism of sheaves of O -modules, then for every integralclosed Z ⊂X, ker(ϕ| ) is
X Z
locally free.
2.2. Diagonalizable derived object.
Definition 2.2.1. Let E be a two-term perfect derived object on a DM stack X.
Wesay E is locally diagonalizable if there is an´etale cover {U}of X such that E|U
can be represented by a diagonalizable homomorphism ϕ:E →E where E and E
1 2 1 2
are two locally free sheaves of O -modules.
U
Using Proposition 3.2, [4] (the universality of diagonalization), one checks di-
rectly that the definition does not depend on the local representation of the ob-
ject E. Here it is worth to point out that in Definition 2.1.1, if the requirement
(p )⊂(p ) is removed,then the “diagonalization”thus defined is not a property
i+1 i
of the derived object but merely a property of a presentation of the object E (cf.
Example ??).
3. The Structures of the Tautological Derived Objects
We will set up the notation and derive the local description of the homorphism
that will be the basis of our modular blowing-ups mentioned in the introduction.
In studying M2(Pn,d) together with π∗f∗OPn(k), we will coverit by´etale opens
U →M (Pn,d)andpickdegreekhyperpsurfacesH ⊂Pningeneralpositionsothat
2
foreach[u,C]∈U,wehaveu−1(H)⊂C is asimpledivisorlie inthe smoothlocus
of C. We let Mdiv be the Artin stack of prestable pairs (C,D) where C are nodal
g
curvesandDareeffectivedivisorsonC. Thus,theassignment[u,C]7→(C,u−1(H)
define morphisms
(3.1) M (Pn,d)⊃U −→Mdiv.
2 2
We pick an affine ´etale V →Mdiv so that the U →Mdiv factor through U →V.
2 2
3.1. Basic setup and the sheaf L. Using the (local) morphism(3.1), it suffices
to study a parallel problem of Mw2t (for OPn(k) (and later for P2).
For a fixed (C,D) ∈ Mdiv lying over (C,w) ∈ Mwt, we pick an affine smooth
2 2
chartV →Mdiv containing(C,D);letρ:C →V andD⊂C betheuniversalfamily
2
on V. We find disjoint sections A ,A ,B ⊂C of C/V, disjoint from D. We list this
1 2
package as
(3.2) ρ:C −→V(→Mwt) and D ⊂C; plus A=A +A , B ⊂C.
2 1 2
Again, D⊂C is the universal divisor and A,B ⊂C are auxiliary divisors.
We let
L =OC(D), M=OC(A−B), and OA(A)=OV(A)⊗OC OA,
and consider the two term complex
(3.3) ϕ:ρ L(A)−→ρ O (A)
∗ ∗ A
via the evaluation homomorphism.
MODULAR RESOLUTION OF MODULI SPACE OF STABLE MAPS 5
Basic Assumptions. We assume throughout this section that
A. a =A ∩C and b=B∩C lie in the core curve F of C; in case F is separable,
i i
thena anda lieondifferentgenus-onesubcurvesofF;weassumethata ,a and
1 2 1 2
b are in general position in F.
B.IncaseDissimple,byan´etalebasechange,wecanassumethatD=D +···+D
1 d
is a union of d disjoint sections D of C →V.
i
C. By shrinking V if necessary, A and B holds for all fibers of C →D.
Under the assumptions, we have R1ρ L(A) = 0; ρ L(A) is locally free, and
∗ ∗
Rρ L is quasi-isomorphic to the complex (3.3).
∗
3.2. Separating nodes and conjugate points. We call a node q (resp. an
irreducible component) of C separating if C−q (resp. C−Σ) is disconnected. An
inseparable curve is a curve with no separating node; an inseparable component of
C is a smallest (via inclusion) inseparable subcurve of C.
Two smooth points p and q of an inseparable genus-2 curve C are conjugate to
each other if OC(p+q)∼=ωC; when p=q, we call it a Weierstrass point of C.
Anynodalgenus-2curveC canbeobtainedbyattachingdisjointconnectedtrees
of rational curves at disjoint smooth points of its core F. We call these connected
trees tails. We call the unique irreducible component of a tail that meets F the
pivotal componentofthe tail. Arationaltailmaycontainseveralchainsofrational
curves; they share the same pivotal component.
Definition3.2.1. WesaythatarationaltailT (resp. amaximalrationalchain R)
is of Weierstrass if its pivotal component intersects the core F at its Weierstrass
point. We say that two rational tails (resp. two maximal rational chains) are
conjugate if their pivotal components intersect F at two conjugate points of F.
3.3. The first reduction. We let ρ : (C,D) → V, A,B ⊂ C and (C,D)× 0 =
V
(C,D) be as in (3.2), satisfying the basic assumptions.
First, the standard inclusion O ⊂ O (D) provides us the obvious section 1 ∈
C C
Γ(ρ L). For other sections, we let M =O (A−B) and consider the inclusion
∗ C
M(D )=O (D +A−B)−⊂→M(D)=O (D+A−B),
i C i C
and the induced inclusions
η :ρ M(D )−⊂→ρ M(D).
i ∗ i ∗
Because of the basic setups, both are locally free. By Riemann-Roch, ρ M(D ) is
∗ i
invertible and ρ M(D) has rank d. We let
∗
(3.4) ϕ:ρ M(D)−→ρ O (A) and ϕ :ρ M(D )−→ρ O (A)
∗ ∗ A i ∗ i ∗ A
be the evaluation homomorphisms; then ϕ =ϕ◦η . We denote their sum by
i i
(η ):⊕d ρ M(D )−→ρ M(D), and (ϕ ):⊕d ρ M(D )−→ρ O (A).
i i=1 ∗ i ∗ i i=1 ∗ i ∗ A
Lemma 3.3.1. We have (1). ρ L ∼= O ⊕ρ L(−B); (2). ρ L(−B) ∼= kerϕ;
∗ V ∗ ∗
(3). (η ) is an isomorphism, and (4) (ϕ )=ϕ◦(η ). Consequently,
i i i
ρ∗L ∼=OV ⊕ker{(ϕi)}.
Proof. The proofs of (1) and (2) are the same as for the genus-1 case, Lemma 4.10
of [3]; we omit the details. We now prove (3). Since both ρ M(D ) and ρ M(D)
∗ i ∗
arelocallyfree,weonlyneedtoshowthatforanyclosedz ∈V,⊕ dρ M(D )| →
i=1 ∗ i z
6 YIHUANDJUNLI
ρ M(D)| is an isomorphism. By base change, it suffices to show that the tauto-
∗ z
logical homomorphism
d d
η (z): H0 O (D +A−B) −→H0 O (D+A−B)
i Cz i Cz
Mi=1 Mi=1 (cid:0) (cid:1) (cid:0) (cid:1)
is an isomorphism. Because both sides are of equal dimensions, it suffices to show
thatitisinjective. TheinjectivityfollowsfromthatD ∩C ,··· ,D ∩C aredistinct
1 z d z
in C . The item (4) is a tautology. This proves the lemma. (cid:3)
z
Thisprovesthatthehomomorphismϕin(3.4)equivalenttothe homomorphism
(ϕ ), via the isomorphism (η ).
i i
We will call either ϕ or (ϕ ) a structural homomorphism. Note that eachϕ has
i i
the form
(3.5) ϕ =ϕ1⊕ϕ2 :ρ M(D )−→ρ O (A )⊕ρ O (A ).
i i i ∗ i ∗ A1 1 ∗ A2 2
These homomorphisms will be our focus in this section.
3.4. The structural homomorphism. We begin with introducing regular func-
tions associated to node-smoothing. By the deformation theory of nodal curves,
for each separating node q ∈ C there is a regular function ζ ∈ Γ(O ) so that
q V
Σ = {ζ = 0} is the locus where the node q is not smoothed; the divisor Σ is a
q q q
smooth divisor. We will refer ζ as a modular parameter.
q
To proceed, we introduce the notation of a node lies between two points (or
components). We saya(separating)node q separates(orlies between)twosmooth
points x and y ∈C if and x and y lie in different connected components of C−q;
in case we replace y by a connected subcurve B ⊂ C, we say q lies between x and
B if q is not a node of B, and x and B−q lie in different connected components
of C −q. We denote by N (resp. N ) to be the set of nodes lying between
[x,y] [x,B]
x and y (resp. B).
Now back to the family C → V with C = C × 0 the central fiber mentioned
V
before. For 1≤i≤m and 1≤i≤2, we denote
δ =D ∩C and a =A ∩C.
i i s s
We form the products of modular parameters via
(3.6) ζ = ζ , and ζ = ζ .
[δi,as] q [δi] q
q∈NY[δi,as] q∈NY[δi,F]
(Here as always, F ⊂ C is its core curve.) In case N = ∅, we set ζ = 1.
[δi,as] [δi,as]
We fix trivializations
(3.7) ρ∗OAs(As)∼=OV, s=1, 2; and the induced ρ∗OA(A)∼=OV⊕2
throughtout this section.
Proposition 3.4.1. For 1 ≤ i ≤ m and 1 ≤ s ≤ 2, after fixing trivializations
ρ∗M(Di)∼=OV and (3.7), the evaluation homomorphism
(3.8) ϕs :ρ M(D )−→ρ O (A )
i ∗ i ∗ As s
is given by the multiplication
(ϕs =) cs·ζ · :O −→O , cs ∈Γ(O∗).
i i [δi,as] V V i V
Proof. The proof is parallel to that of [3, Prop. 4.13]; thus will be omitted. (cid:3)
MODULAR RESOLUTION OF MODULI SPACE OF STABLE MAPS 7
To get hold of higher order vanishing properties, we will treat the restrictions
of the structural homomorphism ϕ on the core and tails separately in the next
subsequent subsections.
3.5. The case of positive inseparable core. In this Subsection, we assume F
is inseparable and degF ∩D > 0. Since degF ∩D = 1 is covered in Proposition
3.4.1, we assume degF ∩D ≥2.
We order the set D ∩F = {δ ,··· ,δ } so that when degD ∩F ≥ 3, then δ
1 r 1
and δ are not conjugate to each other. As always, then D are so ordered that
2 i
δ =C∩D . We denote
i i
ϕF = ϕ : ρ M(D )−→ρ O (A).
i ∗ i ∗ A
δi∈MD∩F δi∈MD∩F
Proposition3.5.1. SupposeF ofC is inseparable, and degD∩F ≥2. Then there
is a trivialization ⊕ri=1ρ∗M(Di)∼=OV⊕r, together with (3.7), such that ϕF takes the
form
c c ··· c
(3.9) ϕF = 11 12 1r
(cid:20)c21 c22 ··· c2r(cid:21)
with ci ∈ Γ(O∗) (for all i,j) such that det(ci) vanishes at 0 if and only if
j V j 1≤i,j≤2
δ and δ are conjugate to each other.
1 2
Proof. The matrix form (3.9) follows directly from Proposition 3.4.1, using that
ζ =1, for all 1≤i≤r and s=1,2.
[δi,δs]
Next, det(a ) vanishes at 0 if and only if h0(O (δ +δ −b))=1 if and
ij 1≤i,j≤2 F 1 2
only if h0(O (δ +δ )) = h0(O (δ +δ −b))+1 = 2 (because the point b is in
F 1 2 F 1 2
general position). The latter is equivalent to that δ and δ are conjugate to each
1 2
other. (cid:3)
For later use, we remark that when degD∩F ≥3, we can re-arrange (if neces-
sary)suchthatδ andδ arenotconjugatetoeachother. Thus,ϕF isdiagonalizable
1 2
in this case.
3.6. A technical proposition on rational curves. We nowfocus onamaximal
chain R ⊂ C of ismooth rational curves R = R ∪···∪R of C attached to a
1 h
smooth point of the core F ⊂C.
We fix some notations. First, we index the R so that R is attached to F, and
j 1
R is attached to R . We let q =F ∩R , and q =R ∩R . Let ζ be the
j+1 j 1 1 j+1 j j+1 qj
modular function associated with q . We list by the ordering
j
(3.10) D∩R={δ ,δ ,··· ,δ }
1 2 r
such that if we define e(i) be such that δ ∈R , then e is nondecreasing. We let
i e(i)
(3.11) ϕR : ρ M(D )−→ρ O (A)
∗ i ∗ A
δi∈MD∩R
be the restriction homomorphism.
Proposition 3.6.1. Let R be a maximal chain of rational curves in (C,D) as
stated. Then after choosing suitable trivialization δi∈D∩Rρ∗M(Di) ∼= OV⊕|D∩R|
and (3.7), the homomorphism ϕR takes the form L
c ζ c ζ ζ c ζ ζ ζ ···
(3.12) ϕR = 11 [δ1,a1] 12 [δ1,a1] [δ2] 13 [δ1,a1] [δ2] [δ3] ,
(cid:20)c21ζ[δ1,a2] c22ζ[δ1,a2]ζ[δ2] c23ζ[δ1,a2]ζ[δ2]ζ[δ3] ···(cid:21)
8 YIHUANDJUNLI
withc ,c ∈Γ(O∗)andc ∈Γ(O )forallj >1. Further,whenF isinseparable,
11 21 V ij V
c c c c c
rank 11 12 13 (0) = 2; rank 11 12 (0) = 2 if and only if q is not a
(cid:20)c21 c22 c23(cid:21) (cid:20)c21 c22(cid:21) 1
Weierstrass point of F.
Proof. We focus on the maximal chain R ⊂ C of smooth rational curves R =
R ∪···∪R of C attached to a smooth point of the core F ⊂C.
1 h
Let ζ be the modular function associated with q . Since q is separating,
qj j j
C× (ζ =0)=G ∪R
V qj j j
isaunionoftwofamiliesofnodalcurvesover(ζ =0)intersecting(i.e. gluedalong)
qj
asectionofnodesΣ =G ∩R ,suchthatΣ ∩C =q andR ∩C =R ∪···∪R .
j j j j j j j+1 h
We letM be the Artinstackofsemi-stablegenustwonodalcurves. (A nodal
2,ss
curve is semi-stable if its smooth rational subcurves contain at least two nodes of
the nodal curve.) By (successively) contracting rational subcurves that contain
only one nodes ofthe ambientcurves,we obtaina semi-stablizationM −→M .
2 2,ss
We define
Mdiv =M × Mdiv.
2,ss 2,ss Mdiv 2
2
PossiblyaftershrinkingandbasechangingV,wecanassumethatthereisasmooth
chart V → M with X → V its universal family so that the composite V →
2,ss
Mdiv →M lifts to g together with a semi-stable contraction g˜ as shown
2 2,ss
V −−−−→ Mdiv
2
(3.13) g , and g˜:C −→C =X × V.
V
 
 
Vy −−−−→ My2,ss
(Note that g˜ contracts all rational tails of the curves in the family C →V.) We let
D =g˜(D ), which are family of smooth divisors of C →V, and let
i i
r r
D = D , D = D .
R i R i
Xi=1 Xi=1
Let ρ¯: C → V be the second projection of (3.13). Because of our construction,
the divisors R ,··· ,R are contracted under the morphism g˜, and
1 h
(3.14) g˜−1(D )=R +···+R +D
i 1 e(i) i
where e is the index function introduced in (3.10).
h
We let r be the number of points in R ∩D; thus r = r. We denote
i i i=1 i
R= h (r +···+r )R ; then P
i=1 i h i
(3.15)P g˜∗O (D +A−B)=O (R+D +A−B).
C R C R
Then we have
(3.16) ρ O (D +A−B)−⊂→ρ O (R+D +A−B)=ρ¯ O (D +A−B).
∗ C R ∗ C R ∗ C R
Here the identity follows from combining (3.15) and the identities
ρ g˜∗O (D +A−B)=ρ¯ g˜ g˜∗O (D +A−B)=ρ¯ O (D +A−B),
∗ C R ∗ ∗ C R ∗ C R
where the last equality holds since both C and C are smooth, g˜:C →C is a proper
divisorial contraction.
MODULAR RESOLUTION OF MODULI SPACE OF STABLE MAPS 9
As before, we denote a =F ∩A , a=a +a , and b=F ∩B. Since they arein
i i 1 2
general position, we have H1(O (rp +a−b)) = 0. Thus by our Basis Setups on
F 1
V, we have that ρ¯ O (D +A−B) is a free Γ(O )-module. For convenience, we
∗ C R V
denote
M=Γ(ρ O (D +A−B)) and M=Γ(ρ¯ O (D +A−B)).
∗ C R ∗ C R
We now pick a basis of M. Since a and b are in general positions. by a simple
vanishing argument, we conclude that Hj(O (a−b)) = 0 for all j. Hence, the
F
restriction homomorphism
H0(O (rp +a−b))−→H0(O (rp )| )
F 1 F 1 rp1
isanisomorphism. Therefore,wecanpickabasiss ,··· ,s ofH0(O (rp +a−b))
1 r F 1
so that s has vanishing order (r−i) at p .
i 1
Next, we let H =(ζ =0)⊂V, and let H=∪r H . We let
i qi i=1 i
ρ¯ :C :=C× H−→H
H H V
betheprojection;letΣ¯ =g˜(Σ ),whereΣ =G ∩R . Forthesamereason,M :=
i i i i i H
(ρ¯ ) O (D +A−B) is a free Γ(O )-module, and has a basis S¯ ,···S¯
H ∗ CH R CH H,1 H,r
so tha(cid:0)t S¯ lifts to S¯′ (cid:1)∈ Γ(O (D +A−B)(−(r−i)Σ¯ )) such that S¯′ are
H,i H,i CH R i H,i
non-vanishing along Σ¯ .
i
By shrinking V if necessary, we can extend S¯ to S¯ ∈ M so the later (as a
H,i i
free Γ(O )-module) is generated freely by S¯ ,··· ,S¯ . Thus g˜∗S¯ ,··· ,g˜∗S¯ freely
V 1 r 1 r
generates the free Γ(O )-module Γ(O (R+D +A−B)).
V C R
Sublemma. Recallζ = i ζ ,andletS =ζ ·g˜∗S¯. Thenthesubmodule
[[qi]] j=1 [qj] i [[qi]] i
M⊂M is a free rank r Γ(OQ)-module generated by S ,··· ,S .
V 1 r
Proof of Sublemma. We begin with introduction useful convention. We let U ⊂ C
i
be an affine open that contains the generic point ξ of R . We let u ∈ Γ(O ) be
i i i Ui
so that R ∩U =(u =0)⊂U . Since R →H is a family of rational curves and
i i i i i i
has general fibers isomorphic to P1, we can find a rational function v on U ∩R
i i i
sothat v restrictsto generalfibers of R ∩U →H arebirationalmaps to A1. We
i i i i
next let ξ′ be the generic point of Σ¯ . Since g˜(ξ )=ξ′, the field K =O contains
i i i i i ξi
Ki′ =Oξi′ as its subfield. Because of our choice of vi, we have Ki =Ki′(vi).
We letOˆ bethe formalcompletionofO alongξ . ThenOˆ =K [[uˆ ]],where
Ui Ui i Ui i i
uˆ is the imageofu inOˆ . (Foranyf ∈Γ(O ), wedenote by[f]its imageinOˆ
i i Ui V Ui
via the pullback O →O and the completion map.)
V Ui
We next pick an open U′ ⊂ C that contains the generic point ξ′ of Σ¯ . By
i i i
shrinking U if necessary, we can assume g˜(U ) ⊂ U′. We then fix a trivialization
i i i
OU′(DR+A−B)∼=OU′, and form the induced trivialization
i i
(3.17) OUi(R+DR+A−B)∼=g˜∗OUi′(DR+A−B)∼=OUi.
Using this trivialization, we can identify any section in Γ(O (R+D +A−B))
Ui R
with a regular function in Γ(O ), thus obtaining its image in Oˆ . We denote this
Ui Ui
process by
γ ∈Γ(OUi(R+DR+A−B))7→[γ]∈OUi(R+DR+A−B)⊗OUi OˆUi ∼=OˆUi.
10 YIHUANDJUNLI
Note that by our choice of S¯ , (i.e. its vanishing along Σ¯ ,) g˜∗S¯ |U has vanishing
j j j i
order r−j along R ; thus [g˜∗S¯ ] = e (uˆ )r−j, where e 6= 0 ∈ Oˆ = K [[uˆ ]] such
i j j i j Ui i i
that uˆ does not divide e .
i j
We now prove the Sublemma. Let F′ ∈Γ(ρ O (R+D +A−B)). By (3.16),
∗ C R
it lies in M if and only if
(3.18) F′| =0∈Γ(O (R+D +A−B)).
R R R
Let r¯ = r +···+r ; it is the multiplicity of R in R. Since g˜∗S¯ has vanishing
i i h i j
order r−j along R , and since ζ has vanishing order max(0,j−(r−r¯)) along
i [[qj]] i
R , S =ζ ·g˜∗S¯ has vanishing order (r−j)+(j−r+r¯)=r¯ along R . Thus
i j [[qj]] j i i i
S lies in M.
j
WenowprovethatMisgeneratedbyS ’s. LetF ∈M. Using(3.16),wedenote
j
itsimageinΓ(ρ O (R+D +A−B))byF′. Usingthe isomorphismbelow (3.16),
∗ C R
and that M is a free Γ(O )-module generated freely by S¯, F′ can be uniquely
V i
expressed as
F′ =f ·g˜∗S¯ +···+f ·g˜∗S¯ , f ∈Γ(O ).
1 1 r r i V
By (3.18), F′|R =0. In particular, (ui)r¯i divides F′|Ui.
We claimthatforanypositiveintegerα, uα divides F′| ifandonlyiff | has
i Ui j Ui
vanishing order max(0,α−r−j) along R , for all 1≤j ≤r. As the if direction is
i
trivial,weprovetheonlyifpart. TakingtheirrespectiveimagesinOˆ ,uα dividing
Ui i
F′| implies that
Ui
(3.19) (uˆ )α divides [F′]=fˆ ·[g˜∗S¯ ]+···+fˆ ·[g˜∗S¯ ]∈K [[uˆ ]].
i 1 1 r r i i
By replacing v by v−1 if necessary, we can assume that ζˆ = c uˆ v , where c 6=
i i qi i i i i
0∈Ki′. Let αj be the order of which fj|Hi is divisible by ζqi, then fˆj =e′j(ζˆqi)αj,
where e′ 6=0∈K′. (The case f =0 is trivially true and is ignored.)
j i j
Addingthat[g˜∗S¯ ]=e (uˆ )r−j mentionedearlier,(3.19)translatestothat(uˆ )α
j j i i
divides
r r
fˆ ·[g˜∗S¯ ]= e′(c uˆ v )αj ·e (uˆ )r−j = vk (e′cke )(uˆ )k+r−j
j j j i i i j i i j i j i
Xj=1 Xj=1 kX≥0 αXj=k
inK [[uˆ ]]=K′(v )[[u ]]. Asuˆ doesnotdividee′cαje ,thisdivisibilityholdsif(uˆ )α
i i i i i i j i j i
divideseachindividualvk(uˆ )k+r−j, whichholdsif, α≤α +r−j foreachj where
i i j
f 6=0. This proves the claim.
j
Applyingtheclaimtoα=r¯i,weconcludethat(ui)r¯i dividingF′|Ui impliesthat
α ≥ r¯ −r +j. This proves that f /ζ , which is a meromorphic function on
j i j [[qj]]
V, is regular over an open subset of V containing the generic point of H , for each
i
1≤i≤r. Since f /ζ is regularover V−∪ H , by Hartogs Lemma, it is regular
j [[qj]] i i
over V.
ThisprovesthatF liesintheΓ(O )spanofS ,··· ,S . SinceMisarankr free
V 1 r
Γ(O )-modules,itisfreelygeneratedbyS ,··· ,S . ThisprovestheSublemma. (cid:3)
V 1 r
We continue the proof of the Proposition. Let bij = S¯j|A¯i ∈ Γ(ρ∗OAi(Ai) ∼=
Γ(O )). Then under {S ,··· ,S } and (3.7), ϕR takes form
V 1 r
b ζ b ζ ζ b ζ ζ ζ ···
(3.20) B = 11 [δ1] 12 [δ1] [δ2] 13 [δ1] [δ2] [δ3] .
(cid:20)b21ζ[δ1] b22ζ[δ1]ζ[δ2] b23ζ[δ1]ζ[δ2]ζ[δ3] ···(cid:21)