On the ample cone of a rational surface with an anticanonical cycle

27 Pages·2013·0.98 MB·English

by  Robert Friedman

Algebra &
Number
Theory
Volume 7
2013
No. 6
On the ample cone of a rational surface with an
anticanonical cycle
Robert Friedman
msp
ALGEBRAANDNUMBERTHEORY7:6(2013)
dx.doi.org/10.2140/ant.2013.7.1481
msp
On the ample cone of a rational surface
with an anticanonical cycle
Robert Friedman
LetY beasmoothrationalsurface,andlet DbeacycleofrationalcurvesonY
thatisananticanonicaldivisor,i.e.,anelementof|−K |. Looijengastudiedthe
Y
geometryofsuchsurfacesY incase Dhasatmostfivecomponentsandidentified
ageometricallysignificantsubsetRofthedivisorclassesofsquare−2orthogonal
tothecomponentsof D. MotivatedbyrecentworkofGross,Hacking,andKeel
ontheglobalTorellitheoremforpairs(Y,D),weattempttogeneralizesomeof
Looijenga’sresultsincase Dhasmorethanfivecomponents. Inparticular,given
anintegralisometry f of H2(Y)thatpreservestheclassesofthecomponents
of D,weinvestigatetherelationshipbetweentheconditionthat f preservesthe
“generic”ampleconeofY andtheconditionthat f preservestheset R.
Introduction
TheampleconeofadelPezzosurfaceY (orrathertheassociateddualpolyhedron)
was studied classically by, among others, Gosset, Schoute, Kantor, Coble, Todd,
Coxeter,andDuVal. Forabriefhistoricaldiscussion,onecanconsulttheremarks
in [Coxeter 1973, §11.x]. From this point of view, the lines on Y are the main
object ofgeometric interest asthey are thewalls ofthe ample coneor the vertices
of the dual polyhedron. The corresponding root system (in case K2 ≤ 6) only
Y
manifestsitselfgeometricallybyallowingdelPezzosurfaceswithrationaldouble
points or, equivalently, smooth surfaces Y with −K nef and big but not ample.
Y
Thisis explicitly worked outin[DuVal1934]. Onthe otherhand,the rootsystem,
or rather its Weyl group, appears for a smooth del Pezzo surface as a group of
symmetries of the ample cone, a fact which (in a somewhat different guise) was
alreadyknowntoCartan. Perhapstheculminationoftheclassicalsideofthestory
is [Du Val 1937], where the blowup of (cid:80)2 at n ≥9 points is also systematically
considered. Inmoderntimes,ManinexplainedtheappearanceoftheWeylgroup
by noting that the orthogonal complement to K in H2(Y;(cid:90)) is a root lattice (cid:51).
Y
Moreover, given any root of (cid:51), in other words an element β of square −2, there
MSC2010: 14J26.
Keywords: rationalsurface,anticanonicalcycle,exceptionalcurve,amplecone.
1481
1482 Robert Friedman
existsadeformationofY forwhichβ =±[C],whereC isasmoothrationalcurve
ofself-intersection−2. For modernexpositionsof thetheory,seefor examplethe
bookofManin[1986]ortheaccountofDemazure[1980a;1980b;1980c;1980d].
Ingeneral,itseemshardtostudyanarbitraryrationalsurfaceY withoutimposing
some extra conditions. One very natural condition is that −K is effective, i.e.,
Y
that −K = D for an effective divisor D. In case the intersection matrix of D
Y
is negative definite, such pairs (Y,D) arise naturally in the study of minimally
elliptic singularities: the case where D is a smooth elliptic curve corresponds to
the case of simple elliptic singularities, the case where D is a nodal curve or a
cycle of smooth rational curves meeting transversally corresponds to the case of
cusp singularities, and the case where D is reduced but has one component with
a cusp, two components with a tacnode, or three components meeting at a point
correspondstotrianglesingularities. Fromthispointofview,thecasewhere D isa
cycleofrationalcurvesisthemostplentiful. Thesystematicstudyofsuchsurfaces
incasetheintersectionmatrixof D isnegativedefinitedatesbackto[Looijenga
1981]. However, for various technical reasons, most of the results of that paper
areprovedundertheassumptionthatthenumberofcomponentsinthecycleisat
most5. Someofthemain pointsofLooijenga’sseminalpaperare asfollows. Let
R denotethesetofelementsin H2(Y;(cid:90))ofsquare−2thatareorthogonaltothe
componentsof D andthatareoftheform±[C],whereC isasmoothrationalcurve
disjointfrom D,forsomedeformationofthepair(Y,D). Intermsofdeformations
ofsingularities,theset R isrelatedtothepossiblerationaldoublepointsingularities
thatcanariseasdeformationsofthedualcusptothecuspsingularitycorresponding
to D. Looijenganotedthat,ingeneral,thereexistelementsin H2(Y;(cid:90))ofsquare
−2thatareorthogonaltothecomponentsof D butthatdonotliein R. Moreover,
reflectionsinelementsoftheset R givesymmetriesofthe“generic”amplecone
(whichisthesameastheampleconeincasetherearenosmoothrationalcurveson
Y disjointfrom D). Finally,stillundertheassumptionofatmostfivecomponents,
any isometry of H2(Y;(cid:90)) that preserves the positive cone, the classes [D ], and
i
theset R preservesthegenericamplecone.
This paper, which is an attempt to see how much of [Looijenga 1981] can be
generalizedtothecaseofarbitrarilymanycomponents,ismotivatedbyaquestion
raised by the recent work of Gross, Hacking, and Keel [Gross et al. 2013] on,
among other matters, the global Torelli theorem for pairs (Y,D) where D is an
anticanonicalcycleontherationalsurfaceY. Inordertoformulatethistheoremin
afairlygeneralway,onewouldliketocharacterizetheisometries f of H2(Y,(cid:90)),
preservingthe positiveconeand fixingthe classes[D ], whichpreservethe ample
i
cone of Y. It is natural to ask if, at least in the generic case, the condition that
f(R) = R is sufficient. In this paper, we give various criteria on R that insure
that, if an isometry f of H2(Y;(cid:90)) preserves the positive cone, the classes [D ],
i
On the ample cone of a rational surface with an anticanonical cycle 1483
and the set R, then f preserves the generic ample cone. Typically, one needs
a hypothesis that says that R is large. For example, one such hypothesis is that
thereis asubsetof R thatspans anegative definitecodimension-1subspaceof the
orthogonalcomplementtothecomponentsof D. Intheory,atleastundervarious
extra hypotheses, such a result gives a necessary and sufficient condition for an
isometrytopreservethegenericamplecone. Inpractice,however,thedetermination
oftheset R ingeneralisadifficultproblem,whichseemscloseinitscomplexityto
theproblemofdescribingthegenericampleconeofY. Finally,weshowthatsome
assumptionson(Y,D)arenecessarybygivingexampleswhere R=∅,sothatthe
condition that an isometry f preserves R is automatic, and of isometries f such
that f preservesthepositivecone,theclasses[D ],and(vacuously)theset R but f
i
doesnotpreservethegenericamplecone. Wedonotyethaveagoodunderstanding
oftherelationshipbetweenpreservingtheampleconeandpreservingtheset R.
An outline of this paper is as follows. The preliminary Section 1 reviews
standardmethodsforconstructingnefclassesonalgebraicsurfacesandappliesthis
to thestudy ofwhen thenormal surface obtainedby contractinga negativedefinite
anticanonical cycle on a rational surface is projective. In Section 2, we analyze
the ample cone and generic ample cone of a pair (Y,D) and show that the set R
defined by Looijenga is exactly the set of elements β in H2(Y;(cid:90)) of square −2
thatareorthogonaltothecomponentsof D suchthatreflectionaboutβ preserves
the generic ample cone. Much of the material of Section 2 overlaps with results
in[Grossetal.2013],provedtherebysomewhatdifferentmethods. Section3is
devoted to giving various sufficient conditions for an isometry f of H2(Y;(cid:90)) to
preservethegenericamplecone,includingtheonedescribedabove. Section4gives
examplesofpairs(Y,D)satisfyingthesufficientconditionsofSection3wherethe
numberofcomponentsof D andthemultiplicity−D2 arearbitrarilylargeaswell
asexamplesshowingthatsomehypotheseson(Y,D)arenecessary.
Notationandconventions. Weworkover(cid:67). If X isasmoothprojectivesurface
withh1(OX)=h2(OX)=0andα∈ H2(X;(cid:90)),welet Lα denotethecorresponding
holomorphiclinebundle,i.e.,c1(Lα)=α. GivenacurveC ordivisorclassG on X,
we let [C] or [G] denote the corresponding element of H2(X;(cid:90)). Intersection
pairingoncurvesordivisors,oronelementsinthesecondcohomologyofasmooth
surface(viewedasacanonicallyoriented4-manifold),isdenotedbymultiplication.
1. Preliminaries
In this paper, Y denotes a smooth rational surface with −K = D =(cid:80)r D a
Y i=1 i
(reduced)cycleofrationalcurves;i.e.,each D isasmoothrationalcurveand D
i i
meets Di±1 transversally,wherei istakenmodr exceptforr =1,inwhichcase
D = D isanirreduciblenodalcurve. Wenote,however,thatmanyoftheresults
1
1484 Robert Friedman
inthispapercanbegeneralizedtothecasewhere D∈|−K |isnotassumedtobe
Y
acycle. Theintegerr =r(D)iscalledthelengthof D. Anorientationof D isan
orientation of the dual graph (with appropriate modifications in caser =1). We
shallabbreviatethe dataofthesurface Y andtheorientedcycle D by(Y,D)and
refertoitasananticanonicalpair. Iftheintersectionmatrix(D ·D )isnegative
i j
definite,wesaythat(Y,D)isanegativedefiniteanticanonicalpair.
Definition 1.1. An irreducible curve E on Y is an exceptional curve if E ∼=(cid:80)1,
E2 =−1, and E (cid:54)= D for any i. An irreducible curve C on Y is a −2-curve if
i
C ∼=(cid:80)1,C2=−2,andC (cid:54)= D foranyi. Let(cid:49) bethesetofall−2-curvesonY,
i Y
andletW((cid:49) )bethegroupofintegralisometriesof H2(Y;(cid:82))generatedbythe
Y
reflectionsintheclassesintheset(cid:49) .
Y
Definition1.2. Let(cid:51)=(cid:51)(Y,D)⊆H2(Y;(cid:90))betheorthogonalcomplementofthe
latticespannedbytheclasses[D ]. FixingtheidentificationPic0 D∼=(cid:71) defined
i m
bytheorientationofthecycle D,wedefinetheperiodhomomorphismϕ :(cid:51)→(cid:71)
Y m
asfollows: ifα∈(cid:51)and Lα isthecorrespondinglinebundle,thenϕY(α)∈(cid:71)m is
theimageofthelinebundleofmultidegree0on D definedby Lα|D. ClearlyϕY is
ahomomorphism. Theperiodmapisthefunctionthatassociatestothepair(Y,D)
thehomomorphismϕ :(cid:51)→(cid:71) .
Y m
By[Looijenga1981;FriedmanandScattone1986;Friedman1984],wehave:
Theorem 1.3. The period map is surjective. More precisely, given Y as above
andgivenanarbitraryhomomorphismϕ:(cid:51)→(cid:71) ,thereexistsadeformationof
m
thepair(Y,D)overasmoothconnectedbase,whichwecantaketobe((cid:71) )n for
m
somen,suchthatthemonodromyofthefamilyistrivialandthereexistsafiberof
thedeformation,say(Y(cid:48),D(cid:48)),suchthatϕY(cid:48) =ϕ undertheinducedidentificationof
(cid:51)(Y(cid:48),D(cid:48))with(cid:51). (cid:3)
Forfuturereference,werecallsomestandardfactsaboutnegativedefinitecurves
onasurface.
Lemma 1.4. Let X be a smooth projective surface, and let G ,...,G be irre-
1 n
duciblecurveson X suchthattheintersectionmatrix(G ·G )isnegativedefinite.
i j
Let F beaneffectivedivisoron X notnecessarilyreducedorirreducibleandsuch
that,foralli,G isnotacomponentof F.
i
(i) Givenr ∈(cid:82), if (F +(cid:80) r G )·G =0 for all j, thenr ≥0 for all i, and,
i i i i j i
for every subset I of {1,...,n}, if (cid:83) G is a connected curve such that
i∈I i
F·G (cid:54)=0forsome j ∈ I,thenr >0fori ∈ I.
j i
(ii) Givens ,t ∈(cid:82),if[F]+(cid:80) s [G ]=(cid:80) t [G ],then F=0ands =t foralli.
i i i i i i i i i i
Thefollowinggeneralresultisalsowellknown:
On the ample cone of a rational surface with an anticanonical cycle 1485
Proposition 1.5. Let X be a smooth projective surface, and let G ,...,G be
1 n
irreducible curves on X such that the intersection matrix (G ·G ) is negative
i j
(cid:83)
definite. (Wedonot,however,assumethat G isconnected.) Thenthereexistsa
i i
nefandbigdivisor H on X suchthat H·G =0forall j and,ifC isanirreducible
j
curve such that C (cid:54)= G for any j, then H ·C > 0. In fact, the set of nef and
j
big (cid:82)-divisors that are orthogonal to {G ,...,G } is a nonempty open subset
1 n
of{G ,...,G }⊥⊗(cid:82).
1 n
Proof. Fix an ample divisor H on X. Since (G · G ) is negative definite,
0 i j
there exist r ∈ (cid:81) such that (cid:0)(cid:80) r G (cid:1)·G = −(H ·G ) for every j. Hence,
i i i i j 0 j
(H +(cid:80) r G )·G =0. ByLemma1.4,r >0foreveryi. Thereexistsan N >0
0 i i i j i
such that Nr ∈ (cid:90) for all i. Then H = N(H +(cid:80) r G ) is an effective divisor
i 0 i i i
satisfying H ·G =0forall j. IfC isanirreduciblecurvesuchthatC (cid:54)=G for
j j
any j,thenH ·C>0andG ·C≥0foralli. Hence, H·C>0. Inparticular, H isnef.
0 i
Finally, H isbigsince H2=NH·(H +(cid:80) r G )=N(H·H )>0as H isample.
0 i i i 0 0
Toseethefinalstatement,weapplytheaboveargumenttoanample(cid:82)-divisor x
(i.e.,anelementintheinterioroftheamplecone)toseethatx+(cid:80) r G isanefand
i i i
big(cid:82)-divisororthogonalto{G ,...,G }. Asx+(cid:80) r G issimplytheorthogonal
1 n i i i
projection pofxonto{G ,...,G }⊥⊗(cid:82)and p:H2(X;(cid:82))→{G ,...,G }⊥⊗(cid:82)
1 n 1 n
isanopenmap,theimageoftheinterioroftheampleconeof X isthenanonempty
opensubsetof{G ,...,G }⊥⊗(cid:82)consistingofnefandbig(cid:82)-divisorsorthogonal
1 n
to{G ,...,G }. (cid:3)
1 n
Applying the above construction to X =Y and D ,...,D , we can find a nef
1 r
andbigdivisor H suchthat H·D =0forall j andsuchthat,ifC isanirreducible
j
curvesuchthatC (cid:54)= D forany j,then H ·C >0.
j
Proposition 1.6. Let (Y,D) be a negative definite anticanonical pair, and let H
be a nef and big divisor such that H · D =0 for all j and such that, if C is an
j
irreduciblecurvesuchthatC (cid:54)= D forany j,then H ·C >0. Supposeinaddition
j
thatO (H)|D=O ,i.e.,thatϕ ([H])=1. Thenthe D arenotfixedcomponents
Y D Y i
of|H|. Hence,ifY denotesthenormalcomplexsurfaceobtainedbycontracting
the D ,then H inducesanampledivisor H onY and|3H|definesanembedding
i
ofY in(cid:80)N forsome N.
Proof. Considertheexactsequence
0→O (H −D)→O (H)→O →0.
Y Y D
Lookingatthelongexactcohomologysequence,as
H1(Y;O (H −D))= H1(Y;O (H)⊗K )
Y Y Y
isSerredualto H1(Y;O (−H))=0,byRamanujam’svanishingtheorem,there
Y
existsasectionofO (H)thatisnowherevanishingon D,provingthefirststatement.
Y
1486 Robert Friedman
ThesecondfollowsfromtheNakai–Moishezoncriterionandthethirdfromgeneral
resultsonlinearseriesonanticanonicalpairs[Friedman1983]. (cid:3)
Remark1.7. Bythesurjectivityoftheperiodmap(Theorem1.3),forany(Y,D)
anegativedefiniteanticanonicalpairand H anefandbigdivisoron Y suchthat
H·D =0forall j and H·C >0forallcurvesC (cid:54)=D ,thereexistsadeformation
j i
of the pair (Y,D) such that the divisor corresponding to H has trivial restriction
to D. Moregenerally,onecanconsiderdeformationssuchthatϕ ([H])isatorsion
Y
point of (cid:71) . In this case, if Y is the normal surface obtained by contracting D,
m
thenY isprojective. Notethatthisimpliesthatthesetofpairs(Y,D)suchthatY
is projective is Zariski dense in the moduli space. However, as the set of torsion
pointsisnotdensein(cid:71) intheclassicaltopology,thesetofprojectivesurfacesY
m
willnotbedenseintheclassicaltopology.
2. Rootsandnodalclasses
Definition2.1. LetC=C(Y)bethepositiveconeofY,i.e.,
C={x ∈ H2(Y;(cid:82)):x2>0}.
ThenChastwocomponents,andexactlyoneofthem,sayC+=C+(Y),contains
theclassesofampledivisors. Wealsodefine
C+ =C+(Y)={x ∈C+:x·[D ]≥0foralli}.
D D i
LetA(Y)⊆C+⊆ H2(Y;(cid:82))be(theclosureof)theample(nef,Kähler)coneofY
inC+. Bydefinition,A(Y)isclosedinC+ butnotingeneralin H2(Y;(cid:82)).
Definition2.2. Letα∈ H2(Y;(cid:90)),α(cid:54)=0. TheorientedwallWα associatedtoα is
theset{x ∈C+:x·α=0},i.e.,theintersectionofC+ withtheorthogonalspace
toα togetherwiththepreferredhalfspacedefinedbyx·α≥0. IfC isacurveonY,
wewrite WC for W[C]. Astandardresult(see,forexample,[FriedmanandMorgan
1988,II(1.8)])showsthat,if I isasubsetof H2(Y;(cid:90))andthereexistsan N ∈(cid:90)+
such that −N ≤α2<0 for all α∈ I, then the collection of walls {Wα :α∈ I} is
locally finite on C+. Finally, we say that Wα is a face of A(Y) if ∂A(Y)∩Wα
containsanonemptyopensubsetof Wα and x·α≥0forall x ∈A(Y).
Lemma2.3. A(Y)isthesetofallx ∈C+ suchthatx·[D ]≥0,x·[E]≥0forall
i
exceptionalcurves E,andx·[C]≥0forall−2-curvesC. Moreover,ifαistheclass
associatedtoanexceptionalor−2-curve,orα=[D ]forsomei suchthat D2<0,
i i
then Wα isafaceofA(Y). Ifα andβ aretwosuchclasses,Wα =Wβ ⇐⇒α=β.
Proof. Forthefirstclaim,itisenoughtoshowthat,ifG isanirreduciblecurveonY
with G2<0,then G iseither D forsomei,anexceptionalcurve,ora−2-curve.
i
Thisfollowsimmediatelyfromadjunctionsince,ifG(cid:54)=D foranyi,thenG·D≥0
i
On the ample cone of a rational surface with an anticanonical cycle 1487
and −2≤2p (G)−2=G2−G·D<0; hence, p (G)=0 and either G2=−2,
a a
G·D=0,or G2=G·D=−1. Thelasttwostatementsfollowfromtheopenness
statementinProposition1.5andthefactthatnotwodistinctclassesofthetypes
listedabovearemultiplesofeachother. (cid:3)
Asanalternatecharacterizationoftheclassesinthepreviouslemma,wehave
thefollowing:
Lemma2.4. Let H beanefdivisorsuchthat H ·D>0.
(i) If α ∈ H2(Y;(cid:90)) with α2 =α·[K ]=−1, then α·[H]≥0 if and only if α
Y
α
is the class of an effective curve. In particular, the wall W does not pass
throughtheinteriorofA(Y). (See[FriedmanandMorgan1988,p.332]fora
moregeneralstatement.)
(ii) Ifβ∈H2(Y;(cid:90))withβ2=−2,β·[D ]=0foralli,β·[H]≥0,andϕ (β)=1,
i Y
then±β istheclassofaneffectivecurve,andβ iseffectiveifβ·[H]>0.
Hence, the ample cone A(Y) is the set of all x ∈ C+ such that x ·[D ] ≥ 0 and
i
x·α≥0forallclassesα andβ asdescribedin(i)and(ii)above,whereincase(ii)
weassumeinadditionthatβ iseffectiveorequivalentlythatβ·[H]>0forsome
nefdivisor H.
Proof. (i) Clearly, if α is the class of an effective curve, then α·[H] ≥ 0 since
H is nef. Conversely, assume that α2 =α·[K ]=−1 and that α·[H]≥0. By
Y
theRiemann–Rochtheorem,χ(Lα)=1. Hence,either h0(Lα)>0or h2(Lα)>0.
But h2(Lα) = h0(L−α1 ⊗ KY) and [H]·(−α−[D]) < 0 by assumption. Thus,
h0(Lα)>0andhenceα istheclassofaneffectivecurve.
(ii)Asin(i), H·(−β−[D])<0,andhence,h0(L−β1⊗KY)=0. Thus,h2(Lβ)=0.
Suppose that h0(Lβ)=0. Then, by the Riemann–Roch theorem, χ(Lβ)=0 and
hence h1(Lβ) = 0. Hence, h1(L−β1⊗KY)=0. Since ϕY(β) = 1, L±β1|D=OD.
Thus,thereisanexactsequence
0→ L−1⊗O (−D)→ L−1→O →0.
β Y β D
Since H1(L−1⊗K )= H1(L−1⊗O (−D))=0, the map H0(L−1)→ H0(O )
β Y β Y β D
issurjectiveandhence−β istheclassofaneffectivecurve. (cid:3)
Definition 2.5. Let α ∈ H2(Y;(cid:90)). Then α is a numerical exceptional curve if
α2=α·[KY]=−1. Thenumericalexceptionalcurveα iseffectiveif h0(Lα)>0,
i.e.,ifα=[G],where G isaneffectivecurve.
AminorvariationoftheproofofLemma2.4showsthefollowing:
Lemma2.6. Let H beanefandbigdivisorsuchthat H ·G >0forallirreducible
curvesG notequalto D forsomei,andletα beanumericalexceptionalcurve.
i
1488 Robert Friedman
(i) Suppose that [H]·α ≥ 0. Then either [H]·α > 0 and α is effective or
H ·D=[H]·α=0andα isanintegrallinearcombinationofthe[D ].
i
(ii) If (Y,D) is negative definite and α is an integral linear combination of the
[D ], then either some component D is a smooth rational curve of self-
i i
intersection−1or K2 =−1,α= K ,andhenceα isnoteffective.
Y Y
(iii) Ifnocomponent D isasmoothrationalcurveofself-intersection−1,thenα
i
iseffectiveifandonlyif[H]·α>0.
Proof. (i) As in the proof of Lemma 2.4, either α or −α − [D] is the class
of an effective divisor. If −α −[D] is the class of an effective divisor, then
0≤[H]·(−α−[D])≤0,so[H]·α= H·D=0. Inparticular,(Y,D)isnegative
definite. Moreover, if G is an effective divisor with [G]=−α−[D], then every
componentof G isequaltosome D . Hence,[G]andthereforeα=−[G]−[D]
i
areintegrallinearcombinationsofthe[D ].
i
(ii)Supposethatα isanintegrallinearcombinationofthe[D ]butthatno D isa
i i
smooth rational curve of self-intersection −1. We shall show that K2 =−1 and
Y
α= K . Firstsupposethat K2 =−1. Then(cid:76) (cid:90)·[D ]=(cid:90)·[K ]⊕L,where L,
Y Y i i Y
the orthogonal complement of [K ] in (cid:76) (cid:90)·[D ], is even and negative definite.
Y i i
Thus,α=a[K ]+β,witheitherβ =0orβ2≤−2,andα2=−a2+β2. Hence,
Y
ifα2=α·[K ]=−1,theonlypossibilityisβ =0anda=1. Incase K2 <−1,
Y Y
D isreducible,andno D isasmoothrationalcurveofself-intersection−1,then
i
D2 ≤−2 for all i and either D2 ≤−4 for some i or there exist i (cid:54)= j such that
i i
D2= D2=−3. In this case, it is easy to check that, for all integers a such that
i j i
a (cid:54)=0forsomei,(cid:0)(cid:80) a D (cid:1)2<−1. Thiscontradictsα2=−1.
i i i i
(iii) If [H]·α >0, then α is effective by (i). If [H]·α <0, then clearly α is not
effective. Suppose that [H]·α =0; we must show that, again, α is not effective.
Supposethatα=[G]iseffective. Bythehypothesison H,everycomponentof G
isa D forsomei sothatα=(cid:80) a [D ]forsomea ∈(cid:90),a ≥0. Let I ⊆{1,...,r}
i i i i i i
be the set of i such that a >0. Then H ·D =0 for all i ∈ I. If I ={1,...,r},
i i
then (Y,D) is negative definite and we are done by (ii). Otherwise, (cid:83) D is a
i∈I i
unionofchainsofcurveswhosecomponents D satisfy D2≤−2. Itistheneasy
i i
tocheckthatα2<−1inthiscase,acontradiction. Hence,α isnoteffective. (cid:3)
Definition2.7. LetY beagenericsmalldeformationofY,andidentify H2(Y ;(cid:82))
t t
with H2(Y;(cid:82)). DefineA =A (Y)tobetheampleconeA(Y )ofY ,viewed
gen gen t t
asasubsetof H2(Y;(cid:82)).
Lemma2.8. Withnotationasabove,thefollowingaretrue:
(i) Iftheredonotexistany−2-curvesonY,thenA(Y)=A . Moregenerally,
gen
A isthesetofallx ∈C+ suchthatx·[D ]≥0andx·α≥0foralleffective
gen i
numericalexceptionalcurves. Inparticular,A(Y)⊆A .
gen
On the ample cone of a rational surface with an anticanonical cycle 1489
(ii) WehaveA(Y)={x ∈A :x·[C]≥0forall−2-curvesC}.
gen
Proof. LetY beasurfacewithno−2-curves(suchsurfacesexistandaregeneric
by the surjectivity of the period map (Theorem 1.3)). Fix a nef divisor H on Y
with H · D > 0. Then A(Y) is the set of all x ∈ C+ such that x ·[D ] ≥ 0 and
i
x ·[E] ≥ 0 for all exceptional curves E, and this last condition is equivalent to
x ·α ≥0 for all α ∈ H2(Y;(cid:90)) such that α2 =α·[K ]=−1 and α·[H]≥0 by
Y
Lemma2.4. SincethisconditionisindependentofthechoiceofY,becausewecan
choosethedivisor H tobeampleandtovaryinasmalldeformation,thefirstpart
of(i)follows,andtheremainingstatementsareclear. (cid:3)
Infact,theargumentaboveshowsthefollowing:
Lemma 2.9. The set of effective numerical exceptional curves and the set A
gen
arelocallyconstantandhenceareinvariantinaglobaldeformationwithtrivial
monodromyundertheinducedidentifications. (cid:3)
Lemma2.10. IfC isa−2-curveonY,thenthewallWC meetstheinteriorofA ,
gen
andinfact,r (A )=A ,wherer : H2(Y;(cid:82))→ H2(Y;(cid:82))isreflectioninthe
C gen gen C
class[C]. Hence,A(Y)isafundamentaldomainfortheactionofthegroupW((cid:49) )
Y
onA ,whereW((cid:49) )isthegroupgeneratedbythereflectionsintheclassesin
gen Y
theset(cid:49) of−2-curvesonY.
Y
Proof. Clearly,ifr (A )=A ,then WC meetstheinteriorofA . Toseethat
C gen gen gen
r (A )=A ,assumefirstmoregenerallythatβ∈(cid:51)isanyclasswithβ2=−2,
C gen gen
andletrβ bethecorrespondingreflection. Thenrβ permutesthesetofα∈H2(Y;(cid:90))
suchthatα2=α·[K ]=−1butdoesnotnecessarilypreservetheconditionthatαis
Y
effective,i.e.,thatα·[H]≥0forsomenefdivisor H onY with H·D>0. However,
for β = [C], there exists by Proposition 1.5 a nef and big divisor H such that
0
H ·C=0and H·D>0. Hence,[H ]isinvariantunderr ,andsor permutesthe
0 0 C C
setofα∈H2(Y;(cid:90))suchthatα2=α·[K ]=−1andα·[H ]≥0. Thus,r permutes
Y 0 C
thesetofeffectivenumericalexceptionalcurvesandhencethefacesofA sothat
gen
r (A )=A . SinceA(Y)⊆A isgivenbyLemma2.8(ii),thefinalstatement
C gen gen gen
isthenageneralresultinthetheoryofreflectiongroups[Bourbaki1981,V§3]. (cid:3)
Remark2.11. (i)TheargumentforthefirstpartofLemma2.10essentiallyboils
down to the following. Let Y be the normal surface obtained by contracting C.
Thenthereflectionr isthemonodromyassociatedtoagenericsmoothingofthe
C
singularsurfaceY,andtheconeA isinvariantundermonodromy.
gen
(ii) If E is an exceptional curve, then WE is a face of A(Y). For a generic Y
(i.e.,no −2-curves),Lemma2.10thensays thatthesetofexceptionalcurveson Y
is invariant under the reflection group generated by all classes of square −2 that
become the classes of a −2-curve under some specialization. A somewhat more
involvedstatementholdsinthenongenericcase.