UNIVERSITA DEGLI STUDI DI PADOVA
Sede Amministrativa: Universita degli Studi diPadova
Dipartimentodi Matemati a Pura e Appli ata
DOTTORATO DIRICERCAIN MATEMATICA
XV CICLO
Quantum Cohomology of Hilb
2
(P 1
P
1
) and
Enumerative Appli ations
Coordinatore: Ch.mo Prof. Bruno Chiarellotto
Supervisore: Ch.ma Prof.ssa Barbara Fante hi
Dottoranda: Dalide Pontoni
DATA CONSEGNA TESI
31 di embre 2003
Introduzione i
Introdu tion iii
1 Some properties of Hilb 2
(P 1
P 1
) 1
1.1 The Hilberts heme asa quotient . . . . . . . . . . . . . . . . 2
1.2 The Hilberts heme asa blowup . . . . . . . . . . . . . . . . 6
1.3 The a tionofAut(Q) on H . . . . . . . . . . . . . . . . . . . 10
1.4 Divisor lassesofH . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 The lo usof non-redu ed pointsofH . . . . . . . . . . . . 15
1.6 The divisor . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.7 Des riptionof some e e tive urves. . . . . . . . . . . . . . . 20
1.8 Subs hemesin identto a given point . . . . . . . . . . . . . . 25
2 Gromov-Witten Invariants 29 2.1 Modulispa e of stablemaps . . . . . . . . . . . . . . . . . . . 29
2.2 Deformation theoryon M 0;n (H; ) . . . . . . . . . . . . . . . 30
2.3 The virtualfundamental lass . . . . . . . . . . . . . . . . . . 33
2.4 A smoothnessresult . . . . . . . . . . . . . . . . . . . . . . . 37
2.5 The modulispa eM 0;0 (H;(0;0; )) . . . . . . . . . . . . . . . 38
2.6 The modulispa eM 0;0 (H;(1;0; )) . . . . . . . . . . . . . . . 40
2.7 Gromov-Witten Invariants . . . . . . . . . . . . . . . . . . . . 46
2.8 Some invariants . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3 Quantum Cohomology 51 3.1 The BigQuantum CohomologyRing . . . . . . . . . . . . . . 51
3.2 A good Q-basis forA (H) . . . . . . . . . . . . . . . . . . . . 53
3.3 The SmallQuantumCohomologyRing . . . . . . . . . . . . . 55
3.4 A presentationof QH s (H). . . . . . . . . . . . . . . . . . . . 57
3.5 The subringgenerated bythedivisor lasses . . . . . . . . . . 67
4 Enumerativeappli ations 71 4.1 The modulispa eof hyperellipti urvesmapping to Q. . . . 71
4.2 The basi orresponden e . . . . . . . . . . . . . . . . . . . . 73
4.3 The maintheorem . . . . . . . . . . . . . . . . . . . . . . . . 75
Lo s hema diHilbertHilb 2
(P 1
P 1
) parametrizzaisottos hemi hiusi zero
dimensionalidi lunghezza duedi P 1
P 1
e risultaessere lis io, irridu ibile
e4-dimensionale. Inquestatesidiamouna presentazioneespli itadellasua
Coomologia Quantum Pi ola. Inoltre elaboriamo un algoritmo (parziale)
he i permetta di al olarne an he la Coomologia Quantum Grande, pur
non essendoingrado didarneuna presentazioneespli ita.
Entrambe le oomologie quantum sonouna deformazione dell'usualeanello
di oomologia H
(Hilb 2
(P 1
P 1
);Q). Si ottengono aggiungendo oppor-
tune variabili formali e de nendo un prodotto he estende il prodotto
[dell'anello di oomologia stesso.
Per ottenere i suddetti risultati utilizziamo la teoria degli spazi di moduli
di mappe stabili, he sono degli sta k nel senso di Deligne-Mumford. In
parti olare usiamo te ni he tipi he dellateoria delledeformazionioltre he
al olidi lassi fondamentalivirtualipersta kdi Deligne-Mumford. Tutto
ioegiusti atodalfatto hei oeÆ ientidelprodottosonogliinvarianti
di Gromov-Witten dello s hema di Hilbert in esame. In questo aso, essi
hannoun signi ato enumerativo, i.e. ontano il numero di urve razionali
hesoddisfano erte proprietadiintersezione, ome ad esempiopassare per
un ssatonumerodi punti. In parti olare mentre laCoomologiaQuantum
Grande oinvolge gli invarianti orrispondentiad un numero n3 di on-
dizionidiin idenza,perquellaPi ola n=3.
In ne abbiamo dimostrato ome si possano ontare le urve iperellitti he
su P 1
P 1
, di genere g 2 e bi-grado (d
1
;d
2
) ssati, he passano per
un erto numero di punti per mezzo degli invarianti di Gromov-Witten
di Hilb 2
(P 1
P 1
). Quest'ultimo risultato e un'appli azione dei al oli di
oomologiaquantumedestendel'analogorisultatoottenutodaTomGraber
perle urve iperellitti hepiane in[Gr℄.
Riteniamo heilmetodousatopertrovarequestirisultatiabbiaraggiuntoil
suolimitenaturale onlostudiodiHilb 2
(P 1
P 1
). Iltentativo diestenderlo
allo s hema di Hilbert di due punti sul blowup di P 2
in un punto o su P n
sie rivelato ineÆ a e a ausa della piu ompli ata struttura degli spazidi
moduli da prendere in onsiderazione, per i quali non disponiamo di una
buonades rizionegeometri a.
Over the last de ades a great interest in the Quantum Cohomology of a
manifoldhas grown outof the work of physi ists (see [W1 ℄, [W2 ℄), provid-
ing a ri h eld of investigation for mathemati ians. In parti ular given a
smooth omplexproje tivevariety X (ora symple ti manifold), there are
twodi erentobje ts whi h anbe alledQuantumCohomologyofX;these
aretheBigQuantumCohomology ringandtheSmallQuantumCohomology
ring.
The Big Quantum Cohomology ring is a -produ t stru ture on V R ,
whereV =H
(X;Q) andR is apowerseriesring,whi hmakesV R into
a R -algebra and redu esto the upprodu twhen putting all thevariables
to zero. The Small Quantum Cohomology ring is de ned by setting equal
to zerosome oftheformal variables, formore detailssee [F-P℄,[G-P ℄.
The -produ t is de ned in terms of the (genus zero) Gromov-Witten in-
variants of X, i.e. thevirtualnumberof genuszerom-pointedstable maps
: C ! X with pres ribed
[C℄ that meet m general y les on X. We
usetheword\virtual"be ausetheGromov-Witteninvariantsneednothave
enumerativesigni an eingeneral. IntheSmallQuantumCohomologyring
onlythe 3-point Gromov-Witten invariants appear. The quantum produ t
an be shown to be ommutative, asso iative, with unit. From the asso-
iativity relations one gets a system of quadrati equations known as the
WDVV-equations(so namedafter E.Witten, R.Dijkgraaf, H.Verlinde,E.
Verlinde by B. Dubrovin). Kontsevi h and Manin in [K-M℄ remark that,
under good hypotheseson X, the WDVV-systemadmits a uniquesolution
on ea fewstartingdata areknown,and itis infa tvery overdetermined.
QuantumCohomology anbeexpli itly omputedusingvarioustools. When
H
(X;Q) is generated byH 2
(X;Q) the same authors prove the First Re-
onstru tionTheorem: itgivesanalgorithmto ndre ursivelyallthegenus
zero Gromov-Witten invariants from the 2-point invariants by means of
the WDVV-equations. The most famous appli ation is due to Kontsevi h
[Kon℄. He al ulates the number of rational urves of degree d in P 2
go-
ingthrough3d 1 points. He onlyneeds asstartingdatum the numberof
lines through two points. Other examples of omputations exploiting the
WDVV-equations an also befoundin[DF-I ℄.
There are some examples of varieties for whi h the Big and/or the Small
QuantumCohomologyringshavebeen omputed,su hasP ,P P [F-P℄,
theblowupofP 2
inr points[G-P℄,Grassmannians[Ber ℄, agvarieties[CF℄,
rationalsurfa es[C-M ℄,some ompleteinterse tions[B℄,themodulispa eof
stablebundles over Riemann surfa es [Mu℄, some proje tive bundles[Q-R℄
andsome blowupsofproje tivebundles[Ma ℄.
AsmoothvarietyX is alled onvex ifH 1
(P 1
;f
T
X
)=0for allgenuszero
stable maps f : P 1
! X. Convexity ensures that the Gromov-Witten in-
variants are enumerative. Only few of the varieties mentioned above are
non- onvex.
A signi ant exampleof a non- onvex variety is represented bythe Hilbert
s heme Hilb n
(X) of n points on a smooth omplex proje tive surfa e X.
It parametrizes the losed 0-dimensional subs hemes of X of length n; it
is smooth, proje tive, 2n-dimensional. For n = 1;2 it is easy to des ribe;
Hilb 1
(X) is X itself and Hilb 2
(X) is obtainedby blowing upX X along
thediagonalandthentakingthequotientbytheobviouslifteda tionofthe
involution. The ase where n =2, X = P 2
has been studied by Graber in
[Gr℄. The author gives a presentation of the Small Quantum Cohomology
ring of the Hilbert s heme by means of quantum deformationsof the rela-
tionsde ningthe Chow ring A
(Hilb 2
(P 2
);Q). Moreoverhe gets enumera-
tiveresultson thehyperellipti plane urvespassing throughanopportune
number of points by studying the modulispa e of genus zero stable maps
into Hilb 2
(P 2
).
Theaim of thisthesis isto study theQuantumCohomologyof the Hilbert
s hemeHilb 2
(P 1
P 1
)andto givesome enumerativeappli ationsextending
Graber'sresultstothe aseofhyperellipti urvesonP 1
P 1
. Thestru ture
ofthework isthefollowing.
Chapter 1 isdevotedto des ribingtheHilberts heme we areworking on.
In x1.1 we follow the above mentioned onstru tion of the Hilbert s heme
as a quotient by the a tion of an involution and we give the orrespond-
ing presentation of its Chow ring whi h is isomorphi to the ohomology
ring. In x1.2 we prove that Hilb 2
(P 1
P 1
) an be seen as a blowup of the
GrassmannianoflinesinP 3
alongtwolinesandalsointhis asewegivethe
orresponding presentationofits Chowring. In parti ularitturnsoutthat
theChowring isnotgeneratedbythedivisor lasses,butwe needto adda
y le lassin odimensiontwo togeta ompletesetofgenerators. Thenwe
studytheindu eda tionoftheautomorphismgroupofP 1
P 1
. TheHilbert
s heme Hilb 2
(P 1
P 1
) is not homogeneous but only almost-homogeneous,
i.e. it has a nitenumber of orbits forming a strati ation. This property
is good enough to make enumerative geometry on it, as shown inx1.3. In
paragraph1.4weanalysethehomogeneouspartofdegree1oftheChowring
ofHilb 2
(P 1
P 1
). Inthefollowingx1.5wegivethegeneratorsofthee e tive
one,postponingadetaileddes riptionofsome onne tede e tive urvesto
x1.8. Paragraphs 1.6and1.7arededi ated tothedes riptionoftwo spe ial
Finallyinx1.9westudythe y lewhosepointsare losedsubs hemesofdi-
mensionzeroandlength2in identtoagivenpointofP 1
P 1
. Itrepresents
a y le lass in A 2
(Hilb 2
(P 1
P 1
)) whi h willbe of ru ial importan efor
appli ationsinChapter 4.
In Chapter 2 we re all the notion of moduli spa e of stable maps (x2.1)
with a brief review of deformation theory in x2.2. Paragraph 2.3 olle ts
some resultsabout thevirtual fundamental lass of a modulispa e and in
x2.4, x2.5, x2.6 weapplythegeneral theoryto some modulispa esofgenus
zerostablemaps into Hilb 2
(P 1
P 1
). The hapter nisheswiththegeneral
de nitionoftheGromov-Witten invariants(x2.7) andthe al ulation(x2.8)
ofsomeinvariantsontheHilberts heme weareinterestedin. Inparti ular,
we arryoutsomeex ess al ulationsonthemodulispa esmentionedabove
involving theirobstru tion bundles.
Chapter 3 olle ts some of the main results. We re all the de nition of
the Big Quantum Cohomology ring of a non- onvex variety and the te h-
niqueswewanttouseinordertoobtainapresentationofitforHilb 2
(P 1
P 1
)
(x3.1). Thenafter xingthenotationsinx3.2,we onstru ttheSmallQuan-
tum Cohomology ring and give a presentation of it inx3.3 and x3.4. This
ispossibleonlyafter making some expli it omputation of Gromov-Witten
invariants using both te hniques from lassi al enumerative geometry and
theWDVV-equations. We on ludethe hapter restri tingourattentionto
the subalgebraS of the Chow ring generated by the divisors lasses. This
allowsusto writea(partial)algorithm al ulatingre ursivelyallthegenus
zero Gromov-Witten invariants of Hilb 2
(P 1
P 1
) starting from few initial
data. The ideaisto dividetheproblemintotwo parts. Theinvariantswith
all the arguments inthe subring S are known by the FirstRe onstru tion
theorem,onlythose involvingthegenerating y le lassin odimensiontwo
areleft and forthem we usetheWDVV-equations.
Chapter 4presentsourmainresult(theorem4.3.1)whi hsolvestheprob-
lem of ounting the hyperellipti urves of given genus and bi-degree on
P 1
P 1
passing througha ertain numberof points whi h mayalso be hy-
perellipti al onjugated (theorems 4.3.5, 4.3.10). In parti ular in x4.1 we
onstru t aspa eparametrizing maps fromahyperellipti urve toP 1
P 1
with good properties. In x4.2 we prove it is anoni ally isomorphi to the
spa e of stable maps from irredu ible rational urves into Hilb 2
(P 1
P 1
)
with good interse tion properties with the strati ation. This means that
we an redu e anenumerativeproblem inhighergenusto aquestion about
rational urves. Finally ourmain theorem is stated and proved in thelast
paragraph 4.3. It extends the result obtained byGraberin [Gr℄, Theorem
2.7, aswellasits appli ationsto theenumerativeproblem.
The main te hni al di eren es between P 1
P 1
and P 2
are related to the
problem of nding a presentationof theQuantumCohomologyrings, sin e
the Chow ring of P 1
P 1
is not generated by the divisor lasses. As said
parts and using two powerful tools as the First Re onstru tion Theorem
and the WDVV-equations. Moreover the des ription of the e e tive one
ofHilb 2
(P 1
P 1
) ismore ompli ated,and requiresto give two geometri al
des riptionsofthe Hilberts heme. We also needto onsider more e e tive
urves for the al ulation of the initial data of the algorithm omputing
(almost) all the Gromov-Witten invariants. This is be ause the group of
automorphismofP 1
P 1
hasmore orbits. Inparti ular wehaveto be are-
ful about interse tion properties of urves with the indu ed strati ation
(theorem 2.4.5).
We thinkthatthete hniquesweused inthisthesishave rea hedtheirnat-
urallimitsandthey an notbesu essfullyappliedto ndanyenumerative
result for example in the ase where X =P n
;n 3, or Bl
p P
2
. In fa t we
onsideredP n
, andfoundoutthatproblemsarisefromstudyingthe ompo-
nentsofex essdimensionofthemodulispa eofgenuszerostablemapsinto
Hilb 2
(P n
). Instead fortheblowupof P 2
ina point we were notableto nd
a simple geometri al des ription of the e e tive one of the orresponding
Hilbert s heme. Moreover also inthis ase the Chow ring of Hilb 2
(Bl
p P
2
)
is not generated by the divisor lasses. Finally the orbits of the indu ed
a tion of the automorphisms group of Bl
p P
2
give a strati ation with no
good interse tion properties.
A knowledgements: IwouldliketothankProfessorBarbaraFante hifor
havingintrodu ed meto the topi of thisthesis and forthetime shespent
dis ussingwith meaboutit.
Iamgratefulto ProfessorAngeloVistolibe ausehe madepossiblemyvisit
to the Dipartimento di Matemati a at the Universita di Bologna. During
mystayingthere I ouldlearna lotaboutthefas inating world ofsta ks.
Ithank ProfessorBruno Chiarellotto,theCo-ordinator ofmy Ph.D. ourse
at the Universita di Padova, who gave me the opportunity to visit people
fromother universities.
In parti ular I would like to thank all the friends from SISSA and ICTP
whomade thelastmonths ofmyPh.D.unforgettable.
Some properties of
Hilb 2
(P 1
P
1
)
Inthis hapterwewill xnotationsand presentsomeresultsontheHilbert
s heme H:=Hilb 2
(P 1
P 1
) whosepoints arerepresented by0-dimensional
length-2 losed subs hemesZ ofP 1
P 1
. Therearetwo possiblegeometri
des riptions of H, as a desingularization of the se ond symmetri produ t
Sym 2
(P 1
P 1
) (see[Fo ℄)and asablowupof theGrassmannianGrass(2;4)
of lines in P 3
(see x1.2). We will give a des ription of both onstru tions
withthe orrespondingChowrings. Thenwewillstudyhowsomeparti ular
divisorsand e e tive urveson H looklike,sothat we willhave adetailed
pi tureof theambient spa e we aregoing to work on.
Notations and onventions: weworkoverC and we identifythevariety
P 1
P 1
with its imageunder theSegre embeddingP 1
P 1
! P 3
, i.e. the
smoothquadri QinP 3
. WehavetworulingsonQ,ifq
1
;q
2
arethetwopro-
je tionsonP 1
, thenq 1
1
(p)representsthe rst rulingand q 1
2
(p)these ond
one.
We onsider Chow ringswithQ- oeÆ ients. Allthevarietiesunder onsid-
erationinthis hapterhavea ellularde omposition,hen etheirChowrings
are isomorphi to theireven- odimension ohomology rings, [Ful℄ Example
19.1.11. Inparti ular we an identifythem.
Givenave torbundleEwedenotebyP(E)theproje tivebundleProj(SymE)
where E is thesheaf of se tions of E. Geometri ally, pointsof P(E) orre-
spond to hyperplanesinthe bers ofE.
We indi atea non-redu ed 0-dimensionalsubs heme Z of length2 of Q as
apair (p;v) wherep2QisthesupportofZ andv2P(T
Q;p
) isadire tion.
We allita non-redu ed pointof H.
1.1 The Hilbert s heme as a quotient
The following des ription of the Hilbert s heme H is valid for all Hilbert
s hemesof 2points ona smoothvariety[F-G ℄.
LetU be theprodu tQQ, pr
1
;pr
2
thetwo proje tions,
~
U the blowupof
U along the diagonal Æ U. The group Z
2
a ts on U xing Æ, so there is
anindu ed a tionon theblowup
~
U. TheHilberts heme H isthe quotient
s heme
~
U=Z
2
, hen eitissmooth,proje tive,irredu ibleand 4-dimensional.
We havethe followingdiagram:
~
Æ j
- ~
U
-
H
Æ bl j
Æ
?
i -
U bl
?
pr1
-
pr
2 - Q
withi;j thenatural in lusions,bltheblowupmap, thequotient mapand
~
Æ theex eptional divisor.
Remark 1.1.1. Given thequotient map :
~
U ! H=
~
U=Z
2
, we have two
indu edhomomorphisms:
: A
(H) -
(A
(
~
U)) Z
2
A
(
~
U)
:A
(
~
U) -
A
(H)
Theyaresu hthat
=2 id=
. Morepre isely:
: A
(H)
-
A
(H)
-
2
: A
(
~
U)
-
A
(
~
U) Z2
-
+
( )
where :
~
U !
~
U is thenatural involutionde ned by( )= . It
followsthatthemap
j
A
(
~
U) Z
2
isthemultipli ationby2homomorphism.
Note that
is an isomorphism of Q-alge bras whi h does not respe t the
degree:
A 4
(H) deg
H
-
Q
A 4
(
~
U) Z
2
?
deg
~
U
-
Q
2
?
Moreover by proje tion formula
is A
4
(H)-linear, where A 4
(
~
U) is made
into an A 4
(H)-algebravia
.
