Stability Conditions, Wall-crossing and weighted Gromov-Witten Invariants

0.29 MB·English

by  Arend Bayer

STABILITY CONDITIONS, WALL-CROSSING AND WEIGHTED
GROMOV-WITTEN INVARIANTS
ARENDBAYERANDYU.I.MANIN
a` PierreDeligne,ente´moignaged’admiration
9
ABSTRACT. We extend B. Hassett’s theory of weighted stable pointed curves
0
([Has03]) to weighted stable maps. The space of stability conditions is described
0
explicitly, and the wall-crossing phenomenon studied. This can be considered as
2
anon-linear analogofthetheoryofstabilityconditionsinabelianandtriangulated

n categories(cf.[GKR04],[Bri07],[Joy06,Joy07a,Joy07b,Joy08]).
a
We introduce virtual fundamental classes and thus obtain weighted Gromov-
J
Witteninvariants. Weshowthatbyincludinggravitationaldescendants,oneobtains
0
anL-algebraasintroducedin[LM04]asageneralizationofacohomological field
2
theory.

]
G
A
§0. Introduction: Hassett’sstabilityconditions
.
h
t 0.1. Pointedcurves. AnodalcurveC overanalgebraically closedfieldk isaproper
a
nodalreducedone-dimensional schemeoffinitetypeoverthisfieldwhoseonlysingu-
m
laritiesarenodes. ThegenusofC isg := dim H1(C,O ).
[ C
Let S be a finite set. A nodal S-pointed curve C is a system (C,s |i ∈ S) where
i
2
{s }isafamilyofclosednon-singulark-pointsofC,notnecessarilypairwisedistinct.
v i
Theelementiiscalledthelabelofs .
0 i
8 The normalization C of C is a disjoint union of smooth proper curves. Each ir-
5
reducible component of C carries inverse images of some labeled points s and of
7 i
0 singular points of C. Teaken together, these points are called special ones. Instead of
6 passingtothenormalizatioen, wemayconsiderbranches(localirreduciblegerms)ofC
0 passingthroughlabeledorsingularpoints. Theyareinanaturalbijection withspecial
/
h points.
t
a Anodal connected S-pointed curve(C,si)iscalled stable ifsi 6= sj fori 6= j and
m anyofthefollowingthreeequivalent conditions hold:
: (i)Theautomorphism groupof(C,s )isfinite.
v i
i (ii)Eachirreducible component ofC ofgenus 0(resp. 1)supports ≥ 3(resp. ≥ 1)
X
distinctspecial points.
r
a (iii)ThelinebundleωC i∈Ssi iesample.
ThisdefinitionhasastraightforwardextensiontofamiliesofstableS-pointedcurves
(cid:0)P (cid:1)
(cf. below). Thebasicresultstatesthatfamilies ofstableS-pointed curvesofgenusg
form(schematicpointsof)aconnectedsmoothproperoverZDeligne-Mumfordstack
M . It contains an open dense substack M parameterizing irreducible smooth
g,S g,S
curves,andisitscompactification.
2000MathematicsSubjectClassification. Primary14N35,14D22;Secondary53D45,14H10,14E99.
Keywordsandphrases. weightedstablemaps,gravitationaldescendants.
1
2 ARENDBAYERANDYU.I.MANIN
0.2. Weighted stability. Generalizing condition (iii), B. Hassett enriched the theory
byadditional parametersgenerating awholenewfamilyofstabilityconditions, which
leadtonewmodulistacks, representing differentcompactifications ofM .
g,S
Namely,theweightdata onS isafunction A : S → Q,0 < A(i) ≤ 1.S together
withaweightdatawillbecalledaweightedset.
0.2.1. Definition ([Has03]). A connected S-pointed curve (C,s |i ∈ S) is called
i
weightedstable(withrespecttoA)ifthefollowingconditions aresatisfied:
(i)K + A(i)s isanampledivisor, whereK isthecanonical classofC.
C i i C
(ii) For any subset I ⊂ S such that s pairwise coincide for i ∈ I, we have
i
P
A(i) ≤ 1.
i∈I
PClearly,(i)impliesthat2g−2+ A(i) > 0.
i
The usual stability notion corresponds to the case A(i) = 1 for all i ∈ S. Inde-
P
pendently ofHassett’s work,A.LosevandYu.Maninconsidered in[LM00],[LM04]
somenon-standard moduli spaces whichturned outtocorrespond tospecial Hassett’s
stability conditions: see[Has03,section6.4]and[Man04].
Definition0.2.1admitsastraightforward extension tofamilies:
Let U be a scheme, S a finite set, g ≥ 0. An S-pointed nodal curve (or family of
curves)ofgenusg overU consists ofthedata
(π : C → U; s : U → C, i∈ S)
i
where π is a flat proper morphism whose geometric fibers C are nodal S-pointed
t
curvesofgenusg.
ThisfamilyiscalledA-stableiff
(i)K + A(i)s isπ-relatively ample.
π i i
(ii)ForanyI ⊂ S suchthat∩ s 6= ∅,wehave A(i) ≤ 1.
P i∈I i i∈I
0.3. Stacks of weighted stable curves M . ThePfirst main result of [Has03] is a
g,A
proof of the following fact. Fix a weighted set of labels S and a value of genus g.
Thenfamilies ofweighted stable S-pointed curves ofgenus g form (schematic points
of) aconnected smooth proper overZDeligne-Mumford stack M .Therespective
g,A
coarsemodulischemeisprojective overZ.
0.4. Walls and wall-crossing. The further results of Hassett on which we focus in
this introduction concern the geometry of the space of stability conditions governing
thevaryinggeometryofboundaries ofM ([Has03],sec. 5).
g,A
Put
D := {A ∈ RS|0 < A(i) ≤ 1, A(i) > 2−2g}.
g,S
s
X
Wallsarenon-emptyintersectionsofD withcertainhyperplanesindexedbysubsets
g,n
I ⊂ S:
w := {A ∈ D | A(i) = 1}.
I g,S
i∈I
X
Coarsechambersaredefinedasconnected components of
D − w .
g,S I
2<|I|≤[n−3δg,0
STABILITYCONDITIONS,WALL-CROSSINGANDWEIGHTEDGROMOV-WITTENINVARIANTS 3
Finechambersareconnected components of
D − w .
g,S I
2≤|I|≤[n−2δg,0
B.Hassettprovesthefollowingresult:
0.4.1. Proposition. (i) The moduli stack M is constant on each coarse chamber,
g,A
anddiffers fromonecoarsechambertoanother.
(ii)Theuniversal curveC isconstantoneachfinechamber,anddiffersfromone
g,A
finechambertoanother.
Finally, foranypoint A′ belonging toawall,thereexists apoint Ainside aneigh-
boring coarse (resp. fine) chamber at which M (resp. C ) is the same as at
g,A g,A
A′.
0.5. Plan of this paper. Let V be a smooth projective manifold. M. Kontsevich has
defined stacks M (V) of S-pointed stable maps (C → V;s ). The stacks M
g,S i g,S
correspond to the case V = a point. In this paper we generalize Hassett’s stability
conditions toM (V)andstudytheresulting stacks.
g,S
In §1, we define the precise moduli problem and construct its moduli space as a
proper Deligne-Mumford stack. Weshowtheexistence ofbirational contraction mor-
phism for any reduction of the weights; in particular, all moduli spaces of weighted
stablemapsarebirational contractions oftheKontsevichmodulispace.
Weestablish theexistence ofallbasicmorphisms (gluing, changing thetarget, for-
getting sections etc.) between them in §2. Section §3 describes the chamber decom-
positions ofthesetofadmissible weights. andexhibits thereduction morphismsfora
wall-crossing asanexplicitblow-up.
In §4, we postulate a list of basic properties for virtual fundamental classes, and
discuss consequences for the weighted Gromov-Witten invariants. After introducing
the language of weighted graphs in §5, we prove a more complete graph-level list of
properties ofthevirtualfundamental classesin§6.
One motivation of this study was the work by Losev and Manin on painted stable
curves [LM00, LM04, Man04], which constitute a special case of weighted stable
curves. The authors introduced the notion of an L-algebra as an extension of the
notionofacohomological fieldtheoryof[KM94].
Theconstruction ofvirtualfundamental classes intheextended contextofnewsta-
bility conditions allows us to produce Gromov-Witten invariants based on weighted
stablemaps. Includinggravitational descendants, weobtainL-algebrasinthesenseof
[LM04]. WhileweightedGromov-Witteninvariantswithoutgravitational descendants
yieldnothingnew(seeproposition 4.2.1),thecouplingtogravityintheweightedcase
exhibitsanewstructure onquantum cohomology.
In[MM08],theauthorsalreadyconstructedmodulispacesofweightedstablemaps
as an aide in computing the Chow ring of non-weighted stable maps with target Pn.
Independently ofthe present paper, Alexeevand Guystudied the behavior ofgravita-
tionaldescendants forchanges ofweightsin[AG08],assumingthesamedefinitionof
virtualfundamental classesthatwestudyinsections §4and§6.
Anothermotivationisspelleddownbelow.
0.6. Stability conditions in abelian and triangulated categories. Stability condi-
tions have been generally designed to choose a preferred compactification of various
4 ARENDBAYERANDYU.I.MANIN
moduli spaces, typically of vector bundles, or more general coherent sheaves on pro-
jectivemanifolds. Itwasonlyrecentlythattheattentionofalgebraicgeometersshifted
to the families of variable stability conditions and their geometry: see [GKR04],
[Bri07], [Joy06, Joy07a, Joy07b, Joy08], and the references therein. An influential
recent paper by T. Bridgeland [Bri07] was very much stimulated by physics work on
mirrorsymmetry,inparticular, M.Douglas’snotion ofΠ-stability.
In this subsection we will sketch a purely geometric context in which various no-
tions of stability in derived categories of coherent sheaves might be quite useful (see
[Ina02],[Bri02],foraversionofbackground notions).
Namely, consider the problem treated in several papers by A. Bondal, D. Orlov
and others: what can be said about a (smooth projective) manifold V if we know its
bounded derivedcategory ofcoherentsheaves D(V)?
In an important paper [BO01] it was shown that if the canonical sheaf Ω of V or
V
its inverse is ample, then V can be reconstructed up to an isomorphism from D(V).
The strategy of proof is this: the authors show how to detect (up to a shift) classes of
structuresheavesofclosedpointsofV inD(V),thenclassesofinvertiblesheaves,and
finallytoreconstruct thecanonical (oranticanonical) homogeneous coordinate ring.
This result can become dramatically wrong, when Ω±1 is not ample, for example,
V
whenitistrivial. IntheproperCalabi-Yaucasevariousbirational modelsmayleadto
equivalentderivedcategories. Thecompletepictureinthiscaseisfarfrombeingclear.
The proof that worked in the Fano/anti-Fano cases breaks down at the first step: the
classesofstructuresheaves ofclosedpointsofV becomeunrecognizable.
However, the general strategy of the proof could be saved without additional as-
sumptions onΩ ifonecoulddothefollowing:
V
a) Devise a family of appropriate stability conditions C (this is probably already
donein[Bri07]).
b)Prove that various V’s with “the same” D(V) could be reconstructed as moduli
spaces V of appropriately defined C-stable point-like complexes in D(V). The de-
C
formation theory of objects in derived categories is not yet a mature subject, but see
[LO06]forsomerecentdevelopments.
c) obtain a sufficiently detailed description of chamber decomposition and wall-
crossing inthespaceofC’s.
A tentative picture of this type can be glimpsed from the Aspinwall’s sketch
[Asp03]. Locally,thewall-crossing phenomenon hasbeenstudied in[Tod08].
Noticehoweverthatitisnotclearaprioriwhatwouldbethenetoutcomeofsucha
reasoning. Infact,accordingtotherecentpreprint[Ca˘l07],twoCalabi-Yauthreefolds
canhaveequivalent derivedcategories withoutbeingbirationally equivalent.
Onthepositiveside,however,theymusthaveisomorphic motives: cf. [Orl05].
From this perspective, Hassett’s theory and its generalization, discussed inthis pa-
per,canbeperceivedasatoymodelforthemoresophisticatedcaseofthetriangulated
categories. Moreover,variousnotionsofstabilityformapsofcurvesintonontrivialtar-
getspacescouldconceivablybecombinedwithsimilarstabilitynotionsforcomplexes
ofsheaves onthetargetspaceleadingtoaricherstructureofquantum cohomology.
0.7. Acknowledgements. The first author would like to thank Andrew Kresch for
usefulremarksonvirtualfundamental classes.
STABILITYCONDITIONS,WALL-CROSSINGANDWEIGHTEDGROMOV-WITTENINVARIANTS 5
§1. Geometryofmodulispacesofweightedstablemaps
1.1. The moduli problem. Let k be a field of any characteristic, V/k a projective
variety, and β ∈ CH1(V)aneffective one-dimensional class inthe Chowring. LetS
beafinitesetwithweightsA: S → Q∩[0,1], andletg ≥ 0beanygenus.
1.1.1. Definition. A nodal curve of genus g over a scheme T/k is a proper, flat mor-
phism π: C → T of finite type such that for every geometric point Spec η of T, the
fiberoverSpecη isaconnected curveofgenusg withonlynodesassingularities.
Given (g,S,A,β) as above, a prestable map of type (g,A,β) over T is a tuple
(C,π,s,f)whereπ: C → T isanodalcurveofgenusg,s = (s ) isanS-tupleof
i i∈S
sections s : T → C,andf isamapf: C → V withf ([C]) = β,suchthat
i ∗
(1) the image of any section s with positive weight A(i) > 0 lies in the smooth
i
locusofC/T,
(2) foranysubsetI ⊂ S suchthattheintersection s (T)ofthecorrespond-
i∈I i
ingsections isnon-empty, wehave A(i) ≤ 1.
i∈I T
1.1.2.Definition. Astablemapoftype(g,A,β)overT isaprestablemap(C,π,s,f)
P
ofthesametypesuchthatK + A(i)s +3f∗(M)isπ-relativelyampleforsome
π i∈S i
ampledivisorM onV.
P
We will often call such a curve (g,A)-stable when the homology class β is irrele-
vant.
1.1.3. Remark. Assume that(C,π,s,f) isa(g,A)-prestable mapover T. Thenitis
(g,A)-stable ifandonlyifitis(g,A)-stable overgeometricpointsofT.
Over an algebraically closed field, ampleness of K + A(i)s + 3f∗(M)
π i∈S i
can only fail on irreducible components C that are of genus 0 and get mapped to a
P
point by f. Precisely, if n is the number of inverse images of nodal points in the
C
normalization, thenamplenessisequivalentton + A(i) > 2.
C i:si∈C
In particular, stability can be checked with an arbitrary ample divisor M; if all
P
sections have weight 1 (wewillwrite this asA = 1 ), weighted stability agrees with
S
thedefinitionofastablemapbyKontsevich.
Wecallthedatag,S,A,β admissible, ifβ 6=0or2g−2+ A(i) > 0,andif
i∈S
β is bounded by the characteristic (cf. [BM96, Theorem 3.14]: this means that k has
P
characteristic zero,orthatβ·L < chark forsomeveryamplelinebundleLonV).
1.1.4.Theorem. Givenadmissible datag,S,A,β, letM (V,β)bethecategory of
g,A
stablemapsoftype(g,A,β)andtheirisomorphisms, withthestandardstructureasa
groupoid overschemesoverSpeck.
Thiscategory isaproperalgebraic Deligne-Mumford stackoffinitetype.
The property of being a stack follows from standard arguments. The geometric
properties areproven insection 1.3. Someoftheir proofs are simplified bythe useof
thecontraction morphismfromtheKontsevichmodulispaceM (V,β)tothespace
g,S
ofweightedstablemapsasdiscussed inthenextsection; hencetheirexistence willbe
provedfirst.
1.2. Reduction morphisms for weight changes. If β 6= 0, consider the open and
densesubstack
C (V,β) ⊂ M (V,β)
g,S g,A
6 ARENDBAYERANDYU.I.MANIN
of maps that do not contract any irreducible component of genus zero, and for which
all marked sections are distinct. By some abuse of language we will call C (V,β)
g,S
the “configuration space”. Since any such map is stable regardless of the choice of
weights,C (V,β)doesnotdependonA. EveryM (V,β)isacompactificationof
g,S g,A
C (V,β),andthusallthemodulistacksfordifferentAarebirational. Thefollowing
g,S
proposition gives actual morphisms, provided that the weights are comparable. They
willbeanalyzed inmoredetailin§3.
Consider two weights A,B: S → Q∩[0,1] such that A(i) ≥ B(i) for all i ∈ S;
we will just write A ≥ B from now on. Any (g,A)-stable map is obviously (g,B)-
prestable, but it may not be (g,B)-stable. However, we can stabilize the curve with
respecttoB:
1.2.1. Proposition. If g,S,β,A ≥ B are as above, there is a natural reduction mor-
phism
ρ : M (V,β) → M (V,β).
B,A g,A g,B
Itissurjective andbirational.1 Overanalgebraically closed fieldη, itisgiven byad-
justingtheweightsandthensuccessively contracting all(g,B)-unstable components.
GiventhreeweightdataA ≥ B ≥ C,thereductionmorphismsrespectcomposition:
ρ = ρ ◦ρ .
C,A C,B B,A
Inparticular,everymodulispaceM (V,β)isabirationalcontractionoftheKont-
g,A
sevichmodulispaceM (V,β) = M (V,β).
g,S g,1
S
1.3. Proofs of the geometric properties. Asin the case of (g,A)-stable curves, the
following vanishing result is essential to ensure that all constructions are compatible
withbasechange:
1.3.1. Proposition. [Has03, Proposition 3.3] Let C be a connected nodal curve of
genus g over an algebraically closed field, D an effective divisor supported in the
smoothlocusofC,andLaninvertible sheafwithL ∼= ωk(D)fork >0.
C
1. IfLisnef,andL 6= ω ,thenLhasvanishing highercohomology.
C
2. IfLisnefandhaspositive degree, thenLN isbasepoint freeforN ≥ 2.
3. IfLisample,thenLN isveryamplewhenN ≥ 3.
4. Assume L is nef and has positive degree, and let C′ denote the image of C
under LN with N ≥ 3. Then C′ is a nodal curve with the same arithmetic genus as
C, obtained by collapsing the irreducible components of C on which L has degree
zero. Components on whichL has positive degree are mapped birationally onto their
images.
1.3.2. Stability and geometric points. Wewill firstshow how remark 1.1.3 follows
fromthisproposition: Considerthelinebundle
L =ωk(k A(i)s )⊗f∗(O(M))3k,
C i
i∈S
X
where k is such that all numbers kA(i) are integral. Then by the proposition and the
basechangetheorems,formationofP := Proj(π (LN))commuteswithbasechange.
∗
1Itisanisomorphism over theopen subset U := Cg,S(V,β), whichsatisfiesthefollowingstrong
densityproperty:foranyopensubsetV,thereisnonon-zerosectionf ∈O(V)thatvanishesonU∩V;
inotherwords,thecomplementisnowheredenseanddoesnothaveadditionalnilpotentstructure.
STABILITYCONDITIONS,WALL-CROSSINGANDWEIGHTEDGROMOV-WITTENINVARIANTS 7
By definition, L is relatively ample iff the induced morphism p: C → P is defined
everywhere and an open immersion. By [SGA1, expose´ I, The´ore`me 5.1], this is the
case if and only if p is everywhere defined, radical, flat and unramified. All these
conditions can be checked on geometric fibers (for flatness, this follows from [EGA,
IV,The´ore`me11.3.10],forunramifiedness fromtheconormalsequence).
1.3.3. Reduction morphisms. By Grothendieck’s descent theory, M (V,β) is a
g,A
stack in the e´tale topology, i. e. the Isom functors are sheaves and any e´tale descent
datum is effective. We first show the existence of the natural reduction morphisms
ρ as maps between these abstract stacks. This will enable us to use the results of
B,A
[BM96]onM (V,β)toshorten ourproofs.
g,S
Using the vanishing result 1.3.1, the proof of proposition 1.2.1 is completely anal-
ogous to that of theorem 4.1 in [Has03]: Let B = λA + (1 − λ)B, and let
λ
1 = λ > λ > ··· > λ = 0be afinite setsuch that for allλ 6∈ {λ ,...,λ }, the
0 1 N 0 N
followingcondition holds:
• ThereisnosubsetI ⊂ S suchthat B (i) = 1and B (i) 6= 1. (*)
i∈I λ i∈I 1
We will construct ρ as the composition ρ = ρ ◦ ··· ◦
B,A P B,A B(PλN),B(λN−1)
ρ . Thismeanswecanassumethatthecondition(*)holdsforall0 < λ< 1.
B(λ1),B(λ0)
FixanampledivisorM onV,andfixanaturalnumberksothatkB(i)isaninteger
for all i. Let L be the invertible sheaf L := ωk(k B(i)s ) ⊗f∗(M)3 for any
C i∈S i
(g,A)-stablemapf: C → V overT. Duetocondition(*),itisnef;alsoithaspositive
degree. LetC′ betheimageofC under themapindPuced byLN forsomeN ≥ 3,i.e.
C′ = Proj R where R is the graded sheaf of rings on T given by R = π ((LN)l).
l ∗
Let t: C → C′ be the natural map, and let s′ = t◦s . By the same arguments as in
i i
the non-weighted case, C′ is a nodal curve of genus g, and s′ lie in the smooth locus
i
wheneverB(i) > 0. Byproposition 1.3.1,Lhasvanishinghighercohomology; sothe
formation of π ((LN)l) and hence that of C′ commutes with base change. Over an
∗
algebraicallyclosedfield,thismorphismagreeswiththedescriptionviacontractionof
unstable components. Inparticular, C′ is(b,B)-prestable.
The original f factors via the induced morphism f′: C′ → V. Let L′ be the line
bundle L′ := ωk (k B(i)s ) ⊗ f′∗(M)3. Then t L = L′; hence L′ is ample
C′ i∈S i ∗
and (C′,π′,s′,f′) is a (g,B)-stable map. The induced morphism T → M (V,β)
g,B
P
commuteswithbasechangeandthusyieldsthemapρ betweenstacksasclaimed.
B,A
To prove surjectivity, it is sufficient to show that every (g,B)-stable map (C,s,f)
overanalgebraicallyclosedfieldK istheimageofsome(g,A)-stablemap(C′,s′,f′)
overK. Itisobvious howtoconstruct C′: IfI ⊂ S isasubset ofthelabels such that
condition (2) of definition 1.1.1 is violated for the weight data A, i.e. the marked
points s ,i ∈ I coincide and A(i) > 1, we can attach a copy of P1(K) at this
i i∈I
point, move the marked points to arbitrary but different points on P1, and extend the
P
mapconstantly alongP1.
Birationality(forβ 6= 0)followsfromthefactthatρ isanisomorphismoverthe
B,A
configuration space C (V,β), which is a dense and open subset. The compatibility
g,S
withcomposition followsimmediately oncewehaveshownthethemodulispacesare
separate: thetwomorphismsρ andρ ◦ρ agreeontheconfiguration space.
C,A C,B B,A
1.3.4. Proposition. The diagonal ∆: M (V,β) → M (V,β)×M (V,β) is
g,A g,A g,A
representable, separated andfinite.
8 ARENDBAYERANDYU.I.MANIN
Let (C ,π ,s ,f ) and (C ,π ,s ,f ) be two families of (g,A)-stable maps to
1 1 1 1 2 2 2 2
V over a scheme T. We have to show that Isom((C ,π ,s ,f ),(C ,π ,s ,f )) is
1 1 1 1 2 2 2 2
represented by a scheme finite and separated over T. Since V is projective and β is
bounded by the characteristic, we can use exactly the same argument as in the proof
of [BM96, Lemma 4.2]: one shows that e´tale locally on T, one can extend the set
of labels to S ∪S′ and find additional S′-tuples of sections (s )′ and (s )′, such that
1 2
(C ,π ,s ∪s′)and(C ,π ,s ∪s′)are(g,A∪1 )-stablecurves,andthatthereis
1 1 1 1 2 2 2 2 S′
anaturalclosedimmersion
Isom((C ,π ,s ,f ),(C ,π ,s ,f )) → Isom((C ,π ,(s ,s′)),(C ,π ,(s ,s′))).
1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2
SineM hasarepresentable,separatedandfinitediagonalby[Has03],theclaim
g,A∪1S′
oftheproposition follows.
1.3.5. Existence as Deligne-Mumford stacks. In particular, the diagonal is proper
andthusthemodulistackseparated. AsM (V,β)isproperandthereductionmor-
g,1
S
phismρ : M (V,β) → M (V,β)issurjective, M (V,β)isalsoproper.
A,1 g,S g,A g,A
S
Finally, the existence of a flatcovering of finite type follows with almost the same
argument as the one in [BM96], following Proposition 4.7 there. However, some
changes are required, so we spell it out in detail: We write A = A ∪ 1 for
n {1,...,n}
o
the weight data obtained from A by adding n weights of 1. Let M (V,β) be the
g,An
open substack of M (V,β) where the additional sections of weight one lie in the
g,An
smooth locus of C (V,β) and away from the existing sections (in other words, the
g,A
open substack where the map is already (g,A)-stable after forgetting the additional
sections). Theobviousforgetful map
φ0 : Mo (V,β) → M (V,β)
A,An g,An g,A
issmoothandinparticularflat. LetU0 (V,β)betheopensubstackofMo (V,β)
g,An g,An
where the curve is already (g,A )-stable as a curve. Then for high enough n,
n
the restriction φo | to this substack is surjective. On the other hand,
A,An Ug0,An(V,β)
U0 (V,β)isanopensubstack oftherelativemorphism spaceMor (V,β)(pa-
g,An Mg,An
rameterizing mapsT → M together withamapofthepull-back oftheuniversal
g,An
curve C to V). So a flat presentation of the morphism space induces one for
g,An
M (V,β).
g,A
§2. Elementarymorphisms
2.1. Gluingmorphisms. Asinthenon-weighted case,wecangluecurvesatmarked
points, buttoguarantee thattheresulting curvesareprestable, wehavetoassumethat
bothlabelshaveweight1:
Letg ,S ,A ,β andg ,S ,A ,β beweightdata,suchthattheextensionsg ,S ∪
1 1 1 1 2 2 2 2 i i
{0},A ∪{0 7→ 1},β by an additional label of weight 1 are admissible. Denote by
i i
ev be the evaluation morphisms ev : M (V,β ) → V given by evaluating
0 0 gi,Ai∪{1} i
theadditional section: ev = f ◦s . Similarly, letg,S,A,β beweightdatasuch that
0 0
g,S∪{0,1},A∪{1,1},β isadmissible,andletev ,ev betheadditional evaluation
0 1
morphisms.
STABILITYCONDITIONS,WALL-CROSSINGANDWEIGHTEDGROMOV-WITTENINVARIANTS 9
2.1.1.Proposition. Therearenaturalgluing morphisms
M (V,β )×M (V,β ) × V → M (V,β +β )
g1,A1∪{1} 1 g2,A2∪{1} 2 V×V g1+g2,A1∪A2 1 2
a(cid:0)nd (cid:1)
M (V,β)× V → M (V,β).
g,A∪{1,1} V×V g+1,A
Theproduct over V ×V istaken via the morphism (ev ,ev )respectively (ev ,ev )
0 0 0 1
ontheleft,andthediagonal ∆: V → V ×V ontheright.
There is nothing new to prove here, except to note that the weight of 1 guarantees
that the marked sections (of positive weight) do not meet the additional node on the
gluedcurve.
2.2.Proposition. Letµ: V → W beamorphism,and(g,S,A,β)beadmissibledata
for V, such that (g,S,A,µ (β)) is also admissible. Then there is a natural push-
∗
forward
M (V,β) → M (W,µ (β))
g,A g,A ∗
thatisobtainedbycomposing themapswithµ,followedbystabilization.
Onecouldadapttheproofof[BM96]totheweightedcase;instead,wegiveaproof
analogous totheoneinsection1.3.3.
Let f: C → V be the universal map over M (V,β), let f′ = µ ◦ f be the
g,A
induced map to W, and let M′ be an ample divisor on V′. By the assumptions, the
divisorD′ = K + A(i)s +3f′∗M′haspositivedegree;however,itneednotbe
π i∈S i
nef. HenceweconsiderD = K + A(i)s +3f∗M andD(λ)= λD+(1−λ)D′
P π i∈S i
for0 ≤ λ ≤ 1. Let{λ ,...,λ }bethesetofλforwhichthedegreeofD(λ)iszero
1 N
P
on any irreducible component of C, and let k ,r = 1...N be an integer such that
r
k λ andk A(i),i ∈ S isinteger.
r r r
Then L = ωk1(k A(i)s + k (3f∗Mλ + (1 − λ )3f′∗M′)) is a nef
1 1 i∈S i 1 1 1
invertible sheaf on C for which proposition 1.3.1 applies. Hence C defined by
1
C := ProjR and (RP) = π (L3l)isagain aflatnodal curve ofgenus g, contract-
1 1 1 l ∗ 1
ing all components of C on which L fails to be ample, and f′ factors via a unique
1
morphism f : C → W. We proceed inductively to obtain f : C → W on which
1 1 N N
D′ is ample; this induces the map of moduli stacks. Note that C → C → W is the
N
universalfactorization off′ suchthatf : C → W isa(g,A)-stable map.
N N
2.3.Proposition. Givenadmissibleweightdata(g,S,A,β),let(g,S∪{∗},A∪{a} =
A {∗ 7→ a},β) be the weight data obtained by adding a label {∗} of arbitrary
weighta∈ Q∩[0,1]. Thereisanaturalforgetfulmap
`
φ : M (V,β) → M (V,β)
A,A∪{a} g,A∪{a} g,A
obtainedbyforgettingtheadditionalsectionandstabilization. Ifa= 0,thenφ
A,A∪{0}
istheuniversal curveoverM (V,β).
g,A
Wecanconstruct thismapasthecomposition
φ ◦ρ : M (V,β) → M (V,β) → M (V,β).
A,A∪{0} A∪{0},A∪{a} g,A∪{a} g,A∪{0} g,A
The second morphism φ isthe naive forgetful morphism, as amap is (g,A∪
A,A∪{0}
{0})-stable ifandonlyifitis(g,A)-stable.
10 ARENDBAYERANDYU.I.MANIN
2.4. Proposition. Let S′ S′′ = S be a partition of the set of labels such that
A(S′′) = A(i) ≤ 1. Thenthereisanaturalmap
i∈S′′ `
P M (V,β) → M (V,β).
g,A|S′∪{A(S′′)} g,A
Itis given by setting s = s for all i ∈ S′. Itidentifies M (V,β)
i ∗ g,A|S′∪{∗7→A(S′′)}
withthelocusofM (V,β)wherealls ,i ∈ S′′ agree.
g,A i
2.5. Weightedmarkedgraphs. Agraph wasdefinedin[BM96]asaquadruple τ =
(V ,F ,∂ ,j )ofasetofverticesV ,asetofflagsF ,amorphism∂ : F → V and
τ τ τ τ τ τ τ τ τ
aninvolution j : F → F . Wethink ofagraphintermsofitsgeometric realization:
τ τ τ
itisobtained byidentifying inthedisjoint union [0,1]thepoints0forallflags
f∈Fτ
f attached to the same vertex via v = ∂ (f), and the points 1 for all orbits of j . A
τ τ
`
flagf withj (f)= f iscalledatailofthevertex∂ (f),whereasapair{f,j (f)}for
τ τ τ
f 6= j (f) is called an edge, connecting the (not necessarily distinct) vertices ∂ (f)
τ τ
and∂ (j (f)).
τ τ
GivenaprojectivevarietyV,aweightedmodularV-graphisagraphτ togetherwith
agenus g: V → Z , aweight data A: F → Q∩[0,1] such that A(f) = 1for all
τ ≥0 τ
flagsthatarepartofanedge,andamarkingβ: V → H+(V). Toanyweightedstable
τ 2
mapwecanassociateitsdualgraph: avertexforeveryirreduciblecomponent,anedge
for every node, and a tail for every marked section. Conversely, to every weighted
modular graph we can associate the moduli space of tuples of weighted stable maps
f : C → V of type (g(v),S = {f ∈ F : ∂(f) = v},A| ,β(v)), such that for
v v v τ Sv
every edge {f,f′ = j (f)} connecting the vertices v = ∂ (f) and v′ = ∂ (f), the
τ τ τ
corresponding evaluationmorphismsareidentical: f ◦s = f ◦s . Viagluing,this
v f v′ f′
gives a single weighted stable map f: C → V; if all C are smooth, its dual graph
v
willgivebackτ.
The moduli space M (V,β) corresponds to the one-vertex graphs with the set
g,A
S of tails. The morphisms constructed in this section correspond to elementary mor-
phismsbetweengraphswithoneandtwovertices. Extending thissetofmorphismsto
higher codimension boundary strata, indexed by graphs with more vertices, naturally
leadstoacategoryofweightedstablemarkedgraphs. Wewilladoptthisviewpointin
§5, and show that we get a functor M from the graph category to Deligne-Mumford
stacksoverk.
§3. Birationalbehaviorunderweightchanges
For this section, we will fix g,S,V,β, and analyze more systematically the reduc-
tion morphisms ρ of proposition 1.2.1 for varying weight data A,B. Assume that
A,B
g,V,β aresuchthatM (V,β)isnotempty.
g,A
3.1. Exceptionallocusandreductionmorphismasblow-up.
3.1.1.Proposition. [Has03,Proposition4.5]AssumewehaveweightdataA ≥ B > 0.
Thereductionmorphismρ contractstheboundarydivisorsD givenastheimage
B,A I,J
ofthegluingmorphism
M (V,0)× M (V,β) → M (V,β)
0,A|I∪{1} V g,A|J∪{1} g,A