Rational surfaces with a large group of automorphisms

0.41 MB·English

by  Serge Cantat

RATIONAL SURFACES WITH A LARGE GROUP OF AUTOMORPHISMS
SERGECANTATANDIGORDOLGACHEV
2
1
0
2
ABSTRACT. Weclassifyrational surfaces X forwhich theimageof theautomor-
n
phisms group Aut(X) in the group of linear transformations of the Picard group
a
J Pic(X) isthe largest possible. Thisanswers aquestion raisedby ArthurCoble in

5 1928,andcanberephrasedintermsofperiodicorbitsofbirationalactionsofinfinite
2 Coxetergroups.

]
G
CONTENTS
A
. 1. Introduction 1
h
t 2. Halphensurfaces 6
a
m 3. Coblesurfaces 14
4. Gizatullin’s TheoremandCremonaspecialpointsetsofninepoints 24
[

5. Thegeneralcase 27
2
v 6. Cremonaspecialsetsofn ≥ 10pointslyingonacubiccurve 36
0 7. Nonalgebraically closedfieldsandothersurfaces 43
3
References 46
9
0
.
6
0
1 1. INTRODUCTION
1
v: LetKbeanalgebraicallyclosedfield,andX beaprojectivesurfacedefinedoverK.
i The group of automorphisms Aut(X) acts on the Ne´ron-Severi group of X. This ac-
X
tionpreservestheintersectionformandthecanonicalclassK ,andthereforeprovides
r X
a a morphism from Aut(X) to the group of integral isometries O(K⊥) of the orthogo-
X
nalcomplement K⊥. WhenX isrational, theimagesatisfies furtherconstraints: Itis
X
contained inanexplicit Coxetersubgroup W ofO(K⊥),and W has infiniteindex
X X X
inO(K⊥)assoonastherank ρ(X)oftheNe´ron-Severigroupof X exceeds11.
X
A natural problem is to describe all projective surfaces X for which Aut(X) is
infinite and its image in this orthogonal or Coxeter group is of finite index. When
K is the field of complex numbers, the problem asks for a classification of complex
projective surfaces with maximal possible groups of isotopy classes of holomorphic
diffeomorphisms.
WesolvethisproblemwhenX isarationalsurface. Itturnsoutthatthisisthemost
interesting anddifficultcase. InSection7.2webrieflydiscussothertypesofsurfaces;
onecantreatthembymoreorlessstandard arguments.
1
2 SERGECANTATANDIGORDOLGACHEV
1.1. Automorphismsofrationalsurfaces. LetXbearationalsurfacedefinedoverK.
TheNe´ron-Severi group of X coincides withthe Picardgroup Pic(X); itsrank ρ(X)
is the Picard number of X. We denote by Aut(X)∗ the image of Aut(X) in the or-
thogonal group O(Pic(X)). There are two alternative possibilities for Aut(X) to be
infinite.
The first occurs when the kernel Aut(X)0 of the action of Aut(X) on Pic(X) is
infinite. In this case, Aut(X)0 is a linear algebraic group of positive dimension and
Aut(X)∗ is a finite group (see [25]). All such examples are easy to describe because
thesurface X isobtained fromaminimalrational surface byasequence ofAut(X)0-
equivariant blow-ups. Toricsurfaces provideexamplesofthiskind.
Inthe second case, the group Aut(X)∗ isinfinite, Aut(X)0 is finite, and then X is
obtained from theprojective plane P2 byblowing upasequence ofpoints p ,...,p ,
1 n
withn ≥ 9(see[35]). TheexistenceofsuchaninfinitegroupAut(X)∗imposesdrastic
constraintsonthepointsetP = {p ,...,p }andleadstonicegeometricpropertiesof
1 n
thisset. Thereareclassical examples ofthis kindaswellas veryrecent constructions
(see [20], [3], [31], [40]). Our goal is to classify point sets P for which the group
Aut(X)∗ isthelargestpossible, inasensewhichwenowmakemoreprecise.
1.2. The hyperbolic lattice. Let Z1,n denote the standard odd unimodular lattice of
signature (1,n). Itisgenerated byanorthogonal basis (e ,e ,...,e )with
0 1 n
e2 = 1, and e2 = −1 for i ≥ 1.
0 i
Theorthogonal complementofthevector
k = −3e +(e +···+e )
n 0 1 n
isasublattice E ⊂ Z1,n. AbasisofE isformedbythevectors
n n
α = e −e −e −e , and α = e −e , i = 1,...,n−1.
0 0 1 2 3 i i i+1
Theintersectionmatrix(α ·α )isequaltoΓ −2I ,whereΓ istheincidencematrix
i j n n n
of the graph T from Figure 1. In particular, each class α has self-intersection
2,3,n−3 i
−2anddeterminesaninvolutive isometryof Z1,n by
s : x 7→ x+(x·α )α .
i i i
Bydefinition, theseinvolutions generatetheCoxeter(orWeyl)group W .
n
s s s s s s
1 2 3 4 n−2 n−1
• • • • ... • •
•s0
FIGURE 1. Coxeter-Dynkin diagramof type T2,3,n−3
AUTOMORPHISMSANDCOBLESURFACES 3
1.3. Automorphisms and Coxeter groups. ¿From now on, X is a rational surface
forwhichAut(X)∗ isinfinite. WewriteX astheblow-up ofP2 atnpoints p ,...,p
1 n
withn ≥ 9;someofthemcanbeinfinitelynearpointsand,byconvention, j ≥ iifp
j
isinfinitely nearp . Wenowdescribe knownconstraints onthestructure ofthegroup
i
Aut(X)∗.
A basis (e ,...,e ) of Pic(X) is obtained by taking for e the class of the total
0 n 0
transform ofaline in P2,andfore ,1 ≤ i ≤ n,theclassofthetotal transform ofthe
i
exceptional divisor obtained by blowing up p ; in particular, the Picard number ρ(X)
i
isequalton+1. Thisbasisisorthogonalwithrespecttotheintersectionform: e2 = 1,
0
e2 = −1fori ≥ 1,ande ·e = 0ifi6= j. WecallsuchabasisofPic(X)ageometric
i i j
basis. A geometric basis makes Pic(X) isometric to the lattice Z1,n, by an isometry
whichmapse toe . Underthisisomorphism, thecanonical class
i i
K = −3e +e +···+e
X 0 1 n
ismappedtotheelementk ∈ Z1,n. Wedenoteby(α )thebasisofK⊥corresponding
n i X
to(α )underthisisomorphism, i.e.
i
(1) α = e −e −e −e , α = e −e , ..., α = e −e ,
0 0 1 2 3 1 1 2 n−1 n−1 n
and by s the involutive isometry of Pic(X) which is conjugate to s . By definition,
i i
the group W is the group ofisometries of Pic(X) generated by these ninvolutions;
X
thus,W isisomorphictotheCoxetergroup W . ItisknownthatthegroupW does
X n X
notdependonachoiceofageometric basis(see[17],Theorem 5.2,page27).
ThegroupAut(X)∗actsbyisometriesonPic(X)andpreservesthecanonicalclass.
According to Kantor-Nagata’s theorem (see [27], Theorem XXXIII, [35], p. 283, or
[17], Theorem 5.2), the group Aut(X)∗ is contained in W . Thus, weget a series of
X
inclusions
Aut(X)∗ ⊂W ⊂ O(Pic(X);K ) ⊂ O(Pic(X)),
X X
where O(Pic(X)) is the orthogonal group of Pic(X) with respect to the intersection
form,andO(Pic(X);K )isthestabilizerofthecanonical class K .
X X
1.4. Cremonaspecialpointsets. Whenn ≤ 8,W isafinitegroup. Wesaythatthe
n
point set P := {p ,...,p }is Cremona special if n ≥ 9 and the group Aut(X)∗ has
1 n
finiteindexinW .1 Arational surfaceobtained byblowingupaCremonaspecialset
X
willbecalledCremonaspecial. Inthissense, Cremonaspecialsurfaces withfixedPi-
card numberarerational surfaces withlargest possible discrete automorphism groups
amongallrationalsurfaces withthesamerankofthePicardgroup.
Our goal is to classify Cremona special point sets, a problem that is already men-
tionedbyArthurCobleinhisbook[10],p. 278(seealso[20]and[25]).
Twokinds ofsuch setsareknownsince thebeginning ofthe lastcentury. Theyare
general Halphen setsof 9points andgeneral Coble setsof 10points (see[10]). Brian
Harbourne showed in [24] that, in characteristic p > 0, and for any integer n ≥ 9, a
1Coblecalledsuchsubsetsspecial,sowesomewhatdeviatefromhisterminology.
4 SERGECANTATANDIGORDOLGACHEV
general setofnnonsingular pointsonanirreducible cuspidal cubiccurveisCremona
special; for this, he employed the fact that all such points are p-torsion points in the
grouplawonthesetofnonsingularpointsonthecubiccurve. Whenn =9,Harbourne
setsareparticularcasesofHalphensets.
We discuss the geometry of Halphen and Coble point sets in Sections 2 and 3,
prove that the general ones are indeed Cremona special, and describe precisely what
“general” meansinthis context. Bydefinition, thepoint set isunnodalifitsblow-up
doesnotcontainsmoothrationalcurveswithself-intersection equalto−2(alsocalled
nodal or (−2)-curves). This terminology is borrowed from the theory of Enriques
surfaces, where itisknownthat theisomorphism classes of unnodal surfaces form an
open subset in the moduli space. A surface obtained by blowing up an unnodal set is
called unnodal. Weshow that unnodal sets form anopen Zariski subset inthe variety
ofpoint setsdefiningHalphen andCoblesurfaces, andthatCremonaspecial Halphen
andCoblesurfacesareexactlytheunnodal ones.
Harbourne examples aredefined atthebeginning ofSection 5andare discussed in
Section6.1.
Then,ourmainresultshowsthattheexamples constructed by Halphen, Coble,and
Harbourne exhaust all possibilities of Cremona special point sets. Asacorollary, if a
pointsetP isCremonaspecial, thenitisunnodal.
Main Theorem Let K be an algebraically closed field. Let P be a Cremona special
pointsetinP2. ThenP isunnodal andoneofthefollowing casesoccurs
K
• n= 9andP isaHalphenset;
• n= 10andP isaCobleset;
• n≥ 10,char(K) > 0,andP isaHarbourneset.
Conversely, anysuchunnodal setisCremonaspecial.
Asacorollary, ifarationalsurfaceX isCremonaspecial,then−K or−2K is
X X
effective.
Remark 1.1. a.–Asexplained inSection7,thereisastrongerversion ofthistheorem
which does not assume that K is algebraically closed, but this requires a careful def-
inition of Cremona special point sets. Non rational surfaces are dealt with in Section
7.2.
1.1. b.–WhenW isinfinite,itisZariskidenseintherealalgebraicgroupO(K⊥⊗
X X
R). Thus, anatural question is the following. If X is a rational surface and Aut(X)∗
isinfinite andZariskidense inO(K⊥ ⊗R),doesitfollow thatX isCremonaspecial
X
? In other words, is it possible to generalize our Main Theorem under the weaker
assumption that Aut(X)∗ is infinite and is Zariski dense in O(K⊥ ⊗R) ? Since W
X 9
contains a finite index, free abelian group of rank 8, every Zariski dense subgroup of
W hasfiniteindex. Thus,theproblemconcerns rationalsurfaceswithPicardnumber
9
atleast10.
1.1. c.– There is a notion of Cremona special point sets in projective spaces of
higher dimension and in their Cartesian products. Interesting examples of such sets
AUTOMORPHISMSANDCOBLESURFACES 5
areknown(see[10],[20],[19]). Unfortunately, themethods ofthis paperarespecific
todimension 2anddonotextendtothehigher-dimensional case.
1.5. An action of W on point sets and its periodic orbits. Consider the variety
n
(P2)n and the diagonal action of PGL on it. Consider the GIT-quotient P2 of the
3 n
action. It turns out that the group W acts on P2 by birational transformations; this
n n
Cremonaactionisdescribed inchapterVIof[20].
Let Γ be a subgroup of W . Let (p ,...,p ) be an ordered stable set of distinct
n 1 n
points representing apoint p ∈ P2. LetX bethesurface obtained byblowingupthe
n p
projective plane at p ,...,p ; its isomorphism class depends only on p. The group
1 n
Pic(X )isisomorphictoZ1,n,withanisomorphismdependingonlyonp;wefixsuch
p
anisomorphism, andthecorresponding isomorphism between W andW .
X n
Ifp is contained in the domain of definition of γ and γ(p) = p forall γ in Γ, then
there exists a subgroup Γ′ ⊂ Aut(X ) such that the action of Γ′ on Pic(X ) and the
p p
identification W ∼ W provide an isomorphism Γ′ → Γ. In other words, points
n Xp
p ∈ P2 whicharefixedbythegroupΓcorrespondtorationalsurfacesonwhichΓis
n
representedbyasubgroupofAut(X ).
p
Thus, ourMain TheoremclassifiesperiodicorbitsofthegroupW ,forn ≥ 9, i.e.
n
forinfinite Coxetergroups W . Thisprovides adynamical interpretation ofthe Main
n
Theoremintermsofbirational actionsofCoxetergroups.
Remark 1.2. Thereareotherniceexamplesofalgebraic dynamical systemsforwhich
periodic orbitsarerelatedtotheconstruction ofinteresting geometricobjects.
One of them is given by Thurston’s pull back map. If F : S2 → S2 is a (topo-
logical) orientation preserving branched covering map of the sphere S2 with a finite
post-critical set2 P of cardinality n, one can ask whether F is equivalent to a holo-
F
morphicendomorphism f oftheRiemannsphere P1(C),inthesensethat
F = φ◦f ◦ψ
where φ and ψ are homeomorphisms which are isotopic relatively to P . The map
F
σ defined by Thurston acts on the Teichmu¨ller space of S2 with n-marked points;
F
fixed points of σ correspond to holomorphic structures on the sphere for which F
F
is realized by an endomorphism f. This situation is similar to the one studied here,
withσ in place of W and the Teichmu¨ller space replacing P2. Wereferto [22]for
F n n
aprecisedescription ofThurston’s construction.
Anothersimilarsituation, withthe mapping class group ofa surface Σ(inplace of
W ) acting on the character variety of the fundamental group π (Σ) (in place of P2)
n 1 n
is related to hyperbolic structures on three dimensional manifolds, and to algebraic
solutions ofPainleve´ sixthequation (see[7]andreferences therein).
2Thepost-criticalsetistheunionoftheimagesofthesetofcriticalpointsoffunderpositiveiterations
off.
6 SERGECANTATANDIGORDOLGACHEV
1.6. Acknowledgement. Thanks to A. Chambert-Loir, J.-L. Colliot-The´le`ne, Y. de
Cornulier, M. W. Davis, M. H. Gizatullin, D. Harari, J. Keum, S. Kondo¯, C.T. Mc-
Mullen, V. Nikulin, and G. Prasad for interesting discussions on the topics of this
paper. We thank the thorough referees for their careful reading and their comments
whichallowedustoclarifytheexposition andtocorrectsomeofthearguments ofthe
previousversionofthispaper.
2. HALPHEN SURFACES
In this section, we describe Halphen surfaces, Halphen pencils of genus 1 curves,
and their associated point sets. We then show that unnodal Halphen point sets are
Cremonaspecial. Mostresultsinthissectionareknowntoexperts,butmaybehardto
findintheliterature, andwillbeusedinthefollowingsections.
Weassumesomefamiliaritywiththetheoryofellipticfibrationsoverfieldsofarbi-
trarycharacteristic andreferto[12],ChapterV,forthistopic.
2.1. Halphensurfacesofindexm. Bydefinition,a (−n)-curveonasmoothprojec-
tivesurfaceX isasmoothrationalcurvewithself-intersection −n. Thegenusformula
showsthefollowing.
Lemma2.1. LetX beasmoothprojective surface. Letnandlbepositive integers.
(1) Assume−K isnef. IfEisa(−n)-curve,thenn = 1,orn = 2andE·K =
X X
0.
(2) Assumethatthelinear system|−lK |contains areduced, irreducible curve
X
C with C2 < 0. If E is a(−n)-curve, then n = 1, or n = 2 and E ·C = 0,
orE = C.
A smooth rational projective surface X is a Halphen surface if there exists an
integer m > 0 such that the linear system |−mK | is of dimension 1, has no fixed
X
component, and has no base point. The index of a Halphen surface is the smallest
possible valueforsuchapositiveinteger m.
LetX beaHalphensurface ofindex m. ThenK2 = 0and, bythegenus formula,
X
thelinearsystem |−mK |definesagenus 1fibration f : X → P1,whichiselliptic
X
orquasi-elliptic ifchar(K) = 2or3. Thisfibrationisrelatively minimalinthesense
thatthereisno(−1)-curve contained inafiber.
Proposition 2.2. Let X be a smooth projective rational surface. Let m be a positive
integer. Thefollowingfourproperties areequivalent:
(i) X isaHalphensurfaceofindexm;
(ii) |−K | is nef and contains a curve F such that O (F ) is of order m in
X 0 F0 0
Pic(F );
0
(iii) there exists a relatively minimal elliptic or quasi-elliptic fibration f : X →
P1; it has no multiple fibers when m = 1 and a unique multiple fiber, of
multiplicitym,whenm > 1;
AUTOMORPHISMSANDCOBLESURFACES 7
(iv) there exists an irreducible pencil of curves of degree 3m with 9 base points
of multiplicity m in P2, such that X is the blow-up of the 9 base points and
|−mK |isthepropertransformofthispencil(thebasepointsetmaycontain
X
infinitelynearpoints).
In the proof of (iii)⇒(iv) below, the classification of minimal rational surfaces is
used. RecallthataminimalrationalsurfaceisisomorphictoP2ortooneoftheSegre-
Hirzebruch surfaces F = P(O ⊕O(−n)),3 with n ≥ 0 and n 6= 1. If n = 0, the
n
surface is isomorphic to P1 ×P1 and, if n = 1, the surface is not minimal since it is
isomorphictotheblow-upofP2atonepoint. Foralln≥ 1thereisauniqueirreducible
curveonF withnegativeself-intersection (equalto−n). Itisdefinedbyasectionof
n
theP1-bundleF → P1corresponding tothesurjection O⊕O(−n)→ O(−n)([26],
n
§V.2).
ProofofProposition 2.2. Underassumption (i), Riemann-Roch formula on arational
surfaceandSerre’sDuality
h0(D)+h0(K −D)= h1(D)+ 1D·(D−K )+1
X 2 X
and K2 = 0 imply that h0(−K ) > 0. Let F be an element of the linear system
X X 0
|−K |.
X
Wenowprove(i)⇔(ii). Theexactsequence
(2) 0 → O → O (nF )→ O (nF )→ 0
X X 0 F0 0
togetherwithh1(X,O )= 0,because X isrational, showthat
X
h0(O (nF )) = 1+h0(O (nF )).
X 0 F0 0
Since F is a nef divisor and F2 = 0, the restriction of O (nF ) to each irreducible
0 0 X 0
component of F is an invertible sheaf of degree zero. The curve F is of arithmetic
0 0
genus 1, so we can apply the Riemann-Roch Theorem on F (see [33], Lecture 11)
0
to conclude that h0(O (nF )) > 0 if and only if O (nF ) ∼= O , if and only if
F0 0 F0 0 F0
h0(O (nF )) = 1.
F0 0
Thisshowsthattheindexmcanbecharacterized bytheproperty
m = min{n :h0(O (−nK )) = 2} = min{n :h0(O (nF )) > 0},
X X F0 0
andtheequivalence (i)⇔(ii)followsfromthischaracterization.
(i)⇒(iii)Thepencil|−mK |definesamorphism f :X → P1 withgeneralfiber
X
ofarithmeticgenus1. ThegenericfiberX isanirreduciblecurveofarithmeticgenus
η
1overthefieldK(η)ofrationalfunctionsonthecurveP1. SinceX issmooth,X isa
η
regularcurveover K(η). Itisknownthatitissmoothifchar(K) 6= 2,3,sothatinthis
case f is an elliptic fibration (see [12], Proposition 5.5.1). If it is not smooth, then a
generalfiberoff isanirreduciblecuspidalcurve,sothatf isaquasi-ellipticfibration.
Asexplainedabove(justbeforeProposition 2.2)thefibrationf isrelatively minimal.
3HereweadoptGrothendieck’sdefinitionoftheprojectivebundleassociatedtoalocallyfreesheaf.
8 SERGECANTATANDIGORDOLGACHEV
SinceX isarationalnon-minimal surface, thereexistsa (−1)-curve E onX satis-
fying E ·K = −1. This shows that K is a primitive divisor class, i.e. K is not
X X X
amultiple ofanyother divisorclass. If m = 1, thisimplies thatthere are nomultiple
fibers. If m > 1, this implies that the multiplicity n ofany multiple fibernD divides
m. Since|mF |isamultiplefiberofmultiplicity m,theclassofthedivisor mF −D
0 n 0
isatorsion element inthePicard group of X. SinceX isarational surface, thisclass
mustbetrivial,andweconclude thatf hasauniquemultiplefiber,namelymF .
0
(iii)⇒(iv) Since thefibration f isrelatively minimal, thecanonical class ispropor-
tional to theclass of thefibersof f (see [1], §V.12); inparticular, K2 = 0and −K
X X
isnef.
Letπ : X → Y beabirational morphism toaminimalruledsurface. Suppose that
Y is not isomorphic to P2; then Y is isomorphic to a surface F , n 6= 1. Let E be
n 0
the section of Y with E2 = −n and let E be its proper transform on X. We have
0
E2 ≤ −nand E2 = −nifand only ifπ isanisomorphism inanopen neighborhood
of E . Since −K is nef, Lemma 2.1 shows that n = 0 or n = 2. Assume n = 2.
0 X
Then π is an isomorphism over E , hence it factors through a birational morphism
0
π : X → X ,whereX istheblow-upofF atapoint x 6∈ E . LetX → F bethe
1 1 2 0 1 1
blow-down of the fiber of the ruling F → P1 passing through x. Then we obtain a
2
birational morphism X → F → P2. Assumenowthatn = 0. Themorphism factors
1
through X → X , where X is the blow-up of a point y on F . Then we compose
2 2 0
X → X withthebirational morphism X → P2 whichisgivenbythestereographic
2 2
projection of F onto P2 from the point y. Asa consequence, changing Y and π, we
0
canalwaysassumethatY = P2.
Since K2 = 0, the morphism π : X → P2 is the blow-up of 9 points p ,...,p ,
X 1 9
where some of them may be infinitely near. Since −K is nef, any smooth rational
X
curve has self-intersection ≥ −2. Thisimplies that the setof points {p ,...,p }can
1 9
bewrittenintheform
(3) {p(1),p(2),...,p(a1);...;p(1),p(2),...,p(ak)},
1 1 1 k k k
where the p(1) are points in P2, and p(b+1) is infinitely near, of the first order, to
j j
(b)
the previous point p for j = 1,...,k and b = 1,...,a − 1. Equivalently, the
j j
exceptional curve
E = π−1(p(1))
j j
is a chain of (−2)-curves of length (a −1) with one more (−1)-curve at the end of
j
thechain.
The formula for the canonical class of the blow-up of a nonsingular surface at a
closedpointshowsthat
(4) K = −3e +e +···+e ,
X 0 1 9
where e = c (π∗(O (1)) and e is the divisor class of E , j = 1,...,9. This
0 1 P2 j j
impliesthat
|−mK | = |3me −m(e +···+e )|,
X 0 1 9
AUTOMORPHISMSANDCOBLESURFACES 9
hence theimageofthepencil |−mK |intheplaneisthelinearsystem ofcurvesof
X
(1)
degree 3mwithsingularpointsofmultiplicity matp ,1≤ j ≤ k.
i
(iv)⇒ (i) Let X be the blow-up of the base points of the pencil. The proper trans-
formofthepencilonX isthelinearsystem |3me −m(e +···+e )|. Theformula
0 1 9
for the canonical class on X shows that this system is equal to |−mK |. Since the
X
pencilisirreducible, |−mK |isapencilwithnofixedcomponentandnobasepoint,
X
soX isaHalphensurface. (cid:3)
Remark2.3. Theproofofthepropositionshowsthatthemultiplicitymofthemultiple
fiber mF of the genus one fibration is equal to the order of O (F ) in Pic(F ).
0 F0 0 0
Thisproperty characterizes non-wildfibersofelliptic fibrations (see[12],Proposition
5.1.5). It is a consequence of the vanishing of H1(X,O ). It always holds if the
X
multiplicity isprimetothecharacteristic.
2.2. Halphenpencilsofindexm. Thefollowinglemmaiswell-knownanditsproof
islefttothereader.
Lemma2.4. Letφ :S′ → Sbetheblow-upofapointxonasmoothprojectivesurface
S andletC′ betheproper transform ofacurvepassing throughxwithmultiplicity 1.
ThenOC′(C′+E) ∼= (φ|C′)∗OC(C),whereE = φ−1(x)istheexceptional divisor.
In the plane P2, an irreducible pencil of elliptic curves of degree 3m with 9 base
pointsofmultiplicitymiscalledaHalphenpencilofindexm. IfC isacubiccurve
0
throughthebasepoints,thenC istheimageofacurveF ∈ |−K |;suchacurveis
0 0 X
uniqueifm > 1andmovesinthepencilifm = 1.
Theclassificationoffibersofgenus1fibrationsshowsthatO (F ) 6∼= O implies
F0 0 F0
that F is a reduced divisor of type I in Kodaira’s notation, unless char(K) divides
0 m
m (see [12], Proposition 5.1.8). We further assume that F is irreducible if m > 1;
0
this will be enough for our applications. Thus F is a smooth or nodal curve, unless
0
the characteristic of K divided m in which case it could be a cuspidal curve. Under
this assumption, the restriction of π to F is an isomorphism F ∼= C ; in particular,
0 0 0
(j)
nobasepointp isasingularpointofC .
i 0
Inthenotation ofEquation(3),considerthedivisorclassinPic(C )givenby
0
d= 3h−a p(1)−···−a p(1),
1 1 k k
wherehistheintersection ofC withalineintheplane. SinceO (C ) ∼= O (3h),
0 C0 0 C0
Lemma2.4gives
O (F )∼= O (3e −e −···−e )∼= (π| )∗(O (d)).
F0 0 F0 0 1 9 F0 C0
This implies that O (F ) is of order m in Pic(F ) if and only if d is of order m in
F0 0 0
Pic(C ). If we choose the group law ⊕ on the set C# of regular points of C with a
0 0 0
nonsingular inflection point o as the zero point, then the latter condition is equivalent
to
(1) (1)
(5) a p ⊕···⊕a p = ǫ
1 1 k k m
10 SERGECANTATANDIGORDOLGACHEV
whereǫ isapointoforderminthegroup (C#,⊕).
m 0
This provides a way to construct Halphen pencils and the corresponding Halphen
surfaces(underourassumptionsthat F isirreducible). Startwithanirreducibleplane
0
cubic C , and choose k points p(1),...,p(1) inC# satisfying Equation (5)witha +
0 1 k 0 1
(1) (1)
...+a = 9. Thenblowupthepointsp ,...,p togetherwithinfinitelynearpoints
k 1 k
p(i),i = 2,...,a ,toarrive atarational surface π : X → P2. Then |−K | = |F |,
j j X 0
whereF isthepropertransformofC . Sincethep(1)satisfyEquation(5),O (F )∼=
0 0 i F0 0
(π| )∗O (ǫ )isoforderminPic(F ). SinceF isirreducible, |−mK | = |mF |
F0 C0 m 0 0 X 0
isnef. Consequently, Proposition 2.2showsthatX isaHalphensurface.
2.3. UnnodalHalphensurfaces. ByLemma2.1,Halphensurfacescontainno(−n)-
curveswithn ≥ 3. RecallthataHalphensurfaceisunnodalifithasno(−2)-curves.
Sincea(−2)-curve R satisfiesR·K = 0,itmustbeanirreducible component ofa
X
fiberofthegenus1fibrationf :X → P1. Conversely,allreduciblefibersoff contain
(−2)-curves, because f isarelativelyminimalellipticfibration. Thus X isunnodalif
andonlyifallmembersofthepencil|−mK |areirreducible.
X
In this case all the curves E are (−1)-curves; in particular, there are no infinitely
i
nearpoints intheHalphen set. Also,inthis case, themorphism f : X → P1 isanel-
lipticfibrationbecauseanyquasi-ellipticfibrationonarationalsurfacehasareducible
fiber whose irreducible components are (−2)-curves (this follows easily from [12],
Proposition 5.1.6, see the proof of Theorem 5.6.3). The fibers of f are irreducible
curvesofarithmeticgenus 1. ThisshowsthataHalphensurfaceisunnodalifandonly
ifitarisesfromaHalphenpencilwithirreduciblemembers.
Proposition 2.5. Let X be a Halphen surface of index m. Then X is unnodal if and
onlyifthefollowing conditions aresatisfied.
(i) Thereisnoinfinitely nearpointintheHalphensetP = {p ,...,p };
1 9
(ii) thedivisor classes
−dK +e −e , i 6= j, 0 ≤ 2d ≤ m,
X i j
−dK ±(e −e −e −e ), i< j < k, 0 ≤ 2(3d±1) ≤ 3m,
X 0 i j k
arenoteffective.
Remark 2.6. SinceK = −3e + e ,thetwotypesofdivisorclasses incondition
X 0 i
P
(ii)areequalto
3de −d(e +···+e )+e −e , i 6= j,
0 1 9 i j
3de −d(e +···+e )±(e −e −e −e ), i < j < k.
0 1 9 0 i j k
Example 2.7. When m = 1, the inequalities 2d ≤ m and 2(3d ± 1) ≤ 3m lead
to d = 0 and the conditions are respectively redundant with (i), or exclude triples of
collinearpointsintheset{p ,...,p }.
1 9
When m = 2, the inequality 2d ≤ m reads d = 0 or 1, and we have to exclude a
cubicthrough8pointswithadoublepointatoneofthem. Theinequality2(3d±1) ≤