Contents lists available atScienceDirect
Journal
of
Number
Theory
www.elsevier.com/locate/jnt
Lines
generate
the
Picard
groups
of
certain
Fermat
surfaces
Alex Degtyarev
DepartmentofMathematics,BilkentUniversity,06800Ankara,Turkey
a r t i c l e i n f o a bs t r a c t
Article history:
Received 29 July 2013
Received in revised form 4 June 2014 Accepted 4 July 2014
Available online 19 September 2014 Communicated by Jean-Louis Colliot-Thélène MSC: primary 14J25 secondary 14J05, 14H30 Keywords: Fermat surface Picard group Néron–Severi group Alexander module
We answera question of T. Shiodaandshow that, forany positiveinteger m primeto 6,thePicardgroupoftheFermat surfaceΦmisgeneratedbytheclassesoflinescontainedin Φm.
Afewotherclassesofsurfacesarealsoconsidered.
© 2014ElsevierInc.All rights reserved.
1. Introduction
1.1. Principalresults
AllalgebraicvarietiesinthepaperareoverC.Let m beapositiveinteger,andlet
Φm:=
zm0 + z1m+ z2m+ zm3 = 0⊂ P3
E-mailaddress:degt@fen.bilkent.edu.tr.
http://dx.doi.org/10.1016/j.jnt.2014.07.020
betheFermatsurface.Ifm= 1 (plane)orm= 2 (quadric),thenΦmcontainsinfinitely manylines(meaningtruestraightlinesinP3);otherwise,Φmisknowntocontainexactly 3m2lines.
Since Φm is simply connected, one can identify its Picard group Pic Φm and its Néron–Severi lattice NS (Φm). Citing [1], the Néron–Severi group “. . . is a rather del-icate invariant of arithmetic nature. Perhaps for this reason it usually requires some nontrivialworkbeforeonecandeterminethePicardnumberofagivenvariety,letalone thefull structureof itsNéron–Severigroup.”The Picardgroupsof Fermatsurfaces are relatedto thoseofthemoregeneralDelsarte surfaces (see[15];theyfit intothe frame-work outlined in Section 2.4). Furthermore, continuing the citation, “Combined with themethodbasedontheinductivestructureof Fermatvarieties,thismightleadtothe verificationoftheHodgeconjectureforallFermatvarieties.”
Let Sm ⊂ Pic Φm be the subgroup generated by the classes of the lines contained in Φm.Then,accordingto[14], onehas
Sm⊗ Q = (Pic Φm)⊗ Q if and only if m 4 or g.c.d.(m, 6) = 1. (1.1) This statement is proved by comparing the dimensions of the two spaces, which are computedindependently.Inotherwords,theclassesoflinesgeneratePic Φmrationally, andanaturalquestion,raisedin[1],iswhethertheyalsogeneratethePicardgroupover theintegers.A partial answerto this questionwasgiven in[12], almost 30 years later: theequalityPic Φm= Smholdsforallintegers m primeto 6 intherange5 m 100. This fact is proved by supersingular reduction and a computer aided computation of thediscriminantsof thelatticesinvolved.(Thecasem= 3 isclassical:any nonsingular cubic contains 27 lines, which generate its Picard group. The case m = 4, i.e., that of K3-surfaces, was settled in [10], see also [3] for a slight generalization. The proof suggestedbelowworksforbothcases.)
Theprincipalresultofthepresentpaperisthefollowingtheorem,answeringtheabove questionintheaffirmativeinthegeneralcase.
Theorem1.2. Letm 1 be aninteger suchthat either m 4 org.c.d.(m,6)= 1. Then
Pic Φm= Sm, i.e.,Pic Φm isgenerated bytheclassesof lines.
Since the 3m2 lines in Φm admit a very explicit description (cf. Section 2.4) and onecan easily see how theyintersect (see, e.g., Eq. (6) in [12]), Theorem 1.2 givesus a complete description of Pic Φm = NS (Φm), including the intersection form and the actionoftheautomorphismgroupof Φm.
Inviewof(1.1),Theorem 1.2isanimmediateconsequenceofthefollowingstatement, which isactually proved inthe paper, see Section4.2. (Throughout the paper, we use Tors A fortheZ-torsionofanabeliangroup/module A.)
In the mean time, an interesting generalization, approaching the problem from a different angle, was suggested in [13]. Briefly, Φm can be represented as an m3-fold ramified coveringoftheplane, andonecantryto studyother multipleplanes withthe sameramificationlocus(seeSection2.4andProblem 2.6fordetails).Consideredin[13] arecyclic coveringsofdegreeatmost 50,and,similarto [12],theproofisalsobasedon comparing thediscriminantsofthetwolattices.
The approach developed in the present paper, including the computation of the Alexander module A[α] (seeSection3.3),whichiscrucial forthe proof,applies to Del-sartesurfaces as well.Here,we makeafewfirststepstowardsthis generalizedproblem and workoutanother specialcase,see Theorem 4.18. Intheforthcomingpaper [5],we close the case of cyclic Delsarte surfaces started in [13] and modify part of the proof of Theorem 1.3(see Section4.2)to adaptitto slightlymoregeneraldiagonal Delsarte
surfaces.On theotherhand,numeric experimentsshowthatTheorem 1.3 doesnot ex-tend literally to allDelsarte surfaces: sometimes,the quotient doeshave torsion. Next specialclassestobestudiedinmoredetailswouldprobablybethenonsingularDelsarte surfaces andthosewith A–D–E singularities.
Asyetanotherapplication,weconsideranotherclassofsurfaceswhose Picardgroup is rationallygenerated bylines, see[3]. Letp and q be twosquare freebivariate homo-geneouspolynomials ofdegree m,anddenote
Σp,q:=
p(z0, z1) = q(z2, z3)
⊂ P3.
This nonsingular surfacecontains anobvious set of m2 lines, viz. those connecting the points [z0: z1: 0: 0] and[0: 0: z2: z3], where p(z0,z1)= q(z2,z3)= 0,and wedenote
bySp,q⊂ Pic Σp,q thesubgroupgeneratedbytheclassesoftheselines.
Theorem 1.4.(SeeSection 4.4.)Forany pairp,q asabove, Tors(Pic Σp,q/Sp,q)= 0. Corollary1.5. (SeeSection4.4.)Ifm isprimeandp,q asaboveare sufficientlygeneric, then Pic Σp,q isgenerated bytheclassesof them2 linescontainedin Σp,q.
1.2. Anoutline oftheproof
In Section 2, we reduce the question to the computation of the torsion of the 1-homologyofacertainspace,seeTheorem 2.2.Wealsorecalltheclassicaldescriptionof thelinesin Φmbymeansofaramifiedcoveringoftheplaneand,following[13],describe ageneralizationoftheproblemtoawiderclassofsurfaces.InSection3,wecomputethe so-called Alexander module (orrather Alexander complex)A[m] of theabove covering and its reduced version A[m].¯ The heart of the proof is a tedious computationof the length( ¯A[m]),see Lemma 4.4inSection4;then, Theorem 1.3followsfrom comparing theresulttotheexpectedvaluegivenby[1,14],seeSection4.2.InSection4.3,wework outatoyexample,illustratingthesuggestedlineofattacktothegeneralizedproblem.
2. Preliminaries
2.1. Prerequisites
For thereader’s convenience, we recall, with references to [7], a few necessary facts from algebraic topology. An ultimate reference would be [8]; unfortunately it is only availableinRussian.
Bydefinition, for any topological pair (X,A) we have thefollowing short exact se-quenceof singularchaincomplexes:
0−→ S∗(A)−→ S∗(X)−→ S∗(X, A)−→ 0.
All complexes are free; hence, applying ⊗G or Hom(·,G), we also have short exact sequences of (co-)chain complexes with any coefficient group G. The associated long exactsequences in(co-)homologyare called the (co-)homologyexactsequences of pair (X,A),cf. (3.2)in[7,Chapter III].
Unless specified otherwise, all (co-)homology are with coefficients in Z. The other groups can be computed using the so-called universal coefficient formulas (see, e.g.,
(7.9)and(7.10) in[7,Chapter VI]):forany topologicalspace X,abelian group G,and integer n,therearenaturalsplit(notnaturally)exactsequences
0−→ Hn(X)⊗ G −→ Hn(X; G)−→ Tor Hn−1(X), G−→ 0, 0−→ ExtHn−1(X), G−→ Hn(X; G)−→ HomHn(X), G −→ 0.
(Here,Tor = Tor1andExt = Ext1arethederivedfunctorsinthecategoryofZ-modules.)
Similar statements hold for the relative groups of pairs (X,A). Assuming all groups finitely generated (e.g.,X is afinite CW -complex), a consequenceof thesecond exact sequence istheassertion thatHn(X) is freeifand onlyif so isH
n−1(X); inthis case,
Hn(X)= Hom(Hn(X),Z).
We use the following terminology for various duality isomorphisms in topology of manifolds.LetM beanorientedcompactmanifold,possiblywithboundary,dim M = n, andA⊂ M a‘sufficientlygood’(seetheendofthisparagraph)closedsubset.If∂M =∅,
themultiplicationbythefundamentalclass[M ] establishescanonicalisomorphisms • Hp(M )= H
n−p(M ) (Poincaréduality,[8,Theorem4in§17.3])and • Hp(M,A)= H
n−p(MA) (Poincaré–Lefschetzduality,[8,Theorem14,Exercise 44, andCorollary thereofin§17.9]).
Ingeneral,themultiplication by[M,∂M ] isanisomorphism
Allstatementsareclassicalandwellknown.Forexample,theycanbederivedasspecial casesofProposition 7.2in[7,ChapterVIII],withanextraobservationthat,inallcases consideredinthepaper,M andA areatworstcompactsemialgebraicsets,thusadmitting finite triangulations (see,e.g., [9]);hence, theyareabsolute neighborhood retractsand the Čech cohomologygroupsin[7] canbe replaced with thesingularones. Asanother consequenceof[9],all(co-)homologygroupsinvolvedarefinitely generated.
2.2. Divisors
Consider a smooth projective algebraic surface X. By Poincaré duality H2(X) =
H2(X),wecanregardtheNéron–SeverilatticeNS (X) asasubgroupoftheintersection
index latticeH2(X)/Tors, representinga divisorD ⊂ X byits fundamentalclass [D],
seeSection2.3below.(TheNéron–Severilattice isthegroupofdivisorsmodulonumeric equivalence; thus, weignorethetorsion.) SincePic X = H1(X;O∗
X) andH2(X;OX) is aC-vectorspace,theexponentialexactsequence
H1(X;OX)−→ H1
X;O∗X−→ H2(X)−→ H2(X;OX) (2.1) impliesthatNS (X) isaprimitivesubgroupinH2(X)/Tors.
IfH1(X)= 0,thenH2(X)= Hom(H2(X),Z) istorsionfree(theuniversalcoefficient
formula), andsoisH2(X)= H2(X). SincealsoH1(X;OX)= H0,1(X) istrivialinthis case, from (2.1) we havePic X = NS (X), i.e., we do notneed to distinguish between linear,algebraic,ornumericequivalence ofdivisors.
ConsiderareducedcurveD⊂ X.Topologically,thenormalizationD of D is˜ aclosed surface,andtheprojectionσ:D˜ → D isahomeomorphismoutsideafinite subsetS ⊂ ˜D.
Wehaveisomorphisms H2(D) ∂ −→ H2 D, σ(S)←− Hσ∗ 2( ˜D, S) ˜ ∂ ←− H2( ˜D) = Z · [Di],
where ∂,∂ are˜ the connecting isomorphisms inthe respective exact sequences of pairs and σ∗ isinducedbytherelativehomeomorphism( ˜D,S)˜ → (D,S).(Forthelatter,one can chooseatriangulationof ˜D with respect towhich thepoints ofS are˜ vertices and project this triangulation to D; then, σ would induce an isomorphism of the relative cellularchaincomplexes.)Thus,H2(D)= H2( ˜D) isthefreeabeliangroupgenerated by
thefundamentalclasses[Di] oftheirreduciblecomponentsDiof D (or,equivalently,the fundamentalclasses[ ˜Di] oftheconnectedcomponentsD˜iof ˜D;inviewofthecanonical isomorphism, we do not distinguish [ ˜Di] from [Di]). A similar computation in coho-mology (theabove sequencewith allindices lifted andall arrows reversed) givesus an isomorphism H2(D) = H2( ˜D). Since thegroup H2( ˜D)= H
0( ˜D) (Poincaré duality)is
torsion free,fromtheuniversalcoefficientformulawe havefurther
H2(D) = H2( ˜D) = HomH2( ˜D),Z
(The last identification uses the canonical basis {[Di]}.) Another application of the universalcoefficientformulashowsthatH1(D) isalsofree.(Essentially,weonlyusethe
factthatthesingularlocushasreal codimensionatleasttwo.)
2.3. Imprimitivity viahomology
Asabove,letD⊂ X beareducedcurveinasmoothprojectivesurface X.Denoting byι:D → X theinclusion,let
SD = Imι∗: H2(D)→ H2(X)/ Tors
.
AsexplainedinSection2.2,SD⊂ NS(X) isthesubgroupgeneratedbytheirreducible componentsof D. For convenience, weretain thenotation ι:D → X and SD in the casewhen D = niDi, ni = 0, isa divisorin X (thus identifying D withitssupport
Di).Thefundamental class ofadivisor D is[D]:=
ni[Di].
Theorem 2.2. Let ι:D → X be as above, and assume that H1(X) = 0. Then there are
canonicalisomorphisms Tors H1(X D) = Hom TD, Q/Z, H1(X D)/ Tors = Hom KD, Z, whereTD:= Tors(NS (X)/SD) and KD:= Ker[ι∗:H2(D)→ H2(X)].
Proof. By Poincaré–Lefschetz duality,we have H1(X D) = H3(X,D).Consider the
followingfragment ofthecohomologyexactsequenceofpair(X,D): H2(X)−→ Hι∗ 2(D)−→ Hδ 3(X, D)−→ H3(X). SinceH3(X)= H
1(X)= 0 (Poincaréduality),wehaveacanonicalisomorphism
H1(X D) = Coker ι∗. (2.3)
As explained above, both H2(X) andH2(D) arefree abelian groups and, for both
spaces,wehaveH2(·)= Hom(H
2(·),Z);hence,ι∗= Hom(ι∗,idZ).Theexactsequence
0−→ KD−→ Hin 2(D)
ι∗
−→ H2(X)
canberegardedasafreeresolutionofQ:= H2(X)/SD.ApplyingHom(·,Z),weobtain
acochain complex
computing the derivedfunctors:H0 = Hom(Q,Z), H1 = Ext(Q,Z), Hi = 0 fori 2.
BythedefinitionofH1 andH2,thisgivesusashort exactsequence 0−→ Ext(Q, Z) −→ Coker ι∗−→ HomKD, Z−→ 0. Here,thefirstgroupisfiniteandthelastoneisfree.Hence,
Ext(Q,Z) = Tors Coker ι∗ and HomKD, Z= Coker ι∗/ Tors .
In viewof(2.3),these two isomorphismsprovethetwo statementsof thetheorem.For thefirststatement,oneshouldalsoobservethatExt(Q,Z)= Ext(Tors Q,Z) (aproperty of finitely generated abelian groups), Tors Q = TD (using the fact that NS (X) is
primitiveinH2(S)),andExt(TD,Z)= Hom(TD,Q/Z) (applyHom(TD,·) tothe
exactsequence0→ Z→ Q→ Q/Z→ 0). 2
2.4. Thecovering Φm→ Φ1
Wemakeextensiveuseoftheramifiedcoveringprm:Φm→ Φ:= Φ1 givenby
prm: [z0: z1: z2: z3] →
z0m: z1m: zm2 : z3m
.
Clearly,Φ istheplane{z0+ z1+ z2+ z3= 0},andprmisramifiedovertheunionoffour lines Ri := Φ∩ {zi = 0}, i = 0,1,2,3.The Galois group of prm is (Z/m)3. Assuming thatm 3,the3m2 linesin Φ
m aretheirreduciblecomponentsofthepreimageof the three linesLi:= Φ∩ {z0+ zi = 0},i= 1,2,3.IntroducethedivisorsL:= L1+ L2+ L3,
R := R0+ R1+ R2+ R3,andV := L+ R inΦ.
With afurther generalizationin mind,redenote Φ[m]:= Φm and consider the pull-backs L∗[m] := pr−1m(L∗), R∗[m] := pr−1m(R∗), and V [m] := pr−1m(V ), where ∗ is an appropriatesubscript,possiblyempty.EachRj[m] isaplanesectionofΦ[m],irreducible and reduced:it is the Fermat curve cutoff Φ[m] bythe plane {zj = 0}. Onthe other hand, L[m] also containsa numberof planesections, e.g., those cutoff by {zi = ξzj},
i= j,ξm=−1.Thus,foranysubsetJ ⊂ {0,1,2,3},onehas S V [m] = S L[m] + RJ[m] = S L[m] = Sm, (2.4) where RJ[m]:= j∈JRj[m].
SinceR isagenericconfigurationoffourlinesintheplane Φ,thefundamentalgroup G:= π1(ΦR) equalsZ3,see[11,Lemma intheproofofTheorem 8].SinceG isabelian,
fromtheHurewicztheoremwehaveG= H1(Φ R)= Hom(KR,Z),seeTheorem 2.2.
This grouphasfour canonicalgenerators gj,j = 0,1,2,3,viz. therestrictionsto KR ofthefourgeneratorsofthegroupH2(R)=
jZ· [Rj]∗.Wehaveg0+ g1+ g2+ g3= 0,
Aninterestinggeneralizationoftheoriginalquestionwassuggestedin[13].Given an epimorphismα:G↠ G toafiniteabeliangroup G,denotebypr:Φ[α]→ Φ theminimal resolution ofsingularities ofthe ramified coveringof Φ defined by α. LetL∗[α], R∗[α], andV [α] bethepull-backsin Φ[α] ofL∗,R∗,and V ,respectively.Tobeconsistentwith thepreviousnotation,weregardaninteger m asthequotientprojectionm:G↠ G/mG.
The components of V [α] (including the exceptional divisors)represent some ‘obvious’ elements of NS (Φ[α]).Using (1.1) and thefinite degree map Φ[m] Φ[α] defined by theinclusionKer α⊂ mG,m:=|G|,onehas
S V [α] ⊗ Q =Pic Φ[α]⊗ Q whenever g.c.d.|G|, 6= 1. (2.5) Thus,itisnaturaltoaskwhetherSV [α]= Pic Φ[α],or,notassumingthat|G| isprime to 6,whetherSV [α]⊂ Pic Φ[α] isaprimitivesubgroup.
Problem2.6.(SeeShimadaandTakahashi[13].)WhendoesonehaveTV [α]= 0? Accordingto[13],theanswertothisquestionisintheaffirmativeiftheimage G of α is acyclic groupof order |G| 50. Anotherexample isworked outinSection4.3,see Theorem 4.18:the answer isalso inthe affirmative ifα(gi)= 0 for at least oneof the standardgeneratorsgi,i= 0,1,2,3.
3. TheAlexandermodule
3.1. The fundamental group
ThelinearrangementL+ R⊂ P2iswellknown;sometimesitisreferredtoasCeva-7.
Itsfundamentalgrouphasbeen computedinmanyways and inmanyplaces; however, sincewewillworkwithaparticularpresentationofthisgroup,werepeatthecomputation here.
Wewillusetheaffine coordinatesx:=−z1/z0, y :=−z3/z0 intheplane Φ.Inthese
coordinates,R0becomesthelineatinfinity,andtheothercomponentsofV arethelines
of theform {rxx+ ryy = r} withrx,ry,r∈ {0,1},see Fig. 1.The fundamentalgroup
π1 := π1(Φ V ) is easily computedby theZariski–van Kampen method[11,16]. Since
we usea modified (or rather intermediate) versionof this approach, we outline briefly its proof, using V as a model. (In full detail, the computation using the projection from a singular point is explained, e.g., in [4].) Consider the projection p:Φ P1,
(x,y) → x.ThisprojectionhasfourspecialfibersFa,viz.thoseoverthepointsa∈ Δ:=
{−1,0,1,∞}.(Three ofthese fibersarecomponentsof V ,butthis factisirrelevantfor themoment.)LetF∗ :=Fa,a∈ Δ.Thentherestrictionp:Φ (V ∪ F∗)→ P1 Δ is alocally trivialfibration and, sinceπ2(P1 Δ)= 0 and the fiberis connected,Serre’s
exactsequence(aka long exactsequenceof afibration)givesusashort exactsequence offundamentalgroups
Fig. 1. The divisor V := L + R⊂ Φ. {1} −→ π1(F V ) −→ π1 Φ (V ∪ F∗)−→ π1 P1 Δ−→ {1},
where F is a typical fiber of p, e.g., the one over x = 1
2. Choosing ( 1 2,−
3
2) for the
basepoint,wehaveπ1(F V )=v1,v2,v3,v4,seeFig. 1.Thegroupπ1(P1 Δ) isfree,
and theexactsequence splits. A splittingcanbe constructedgeometrically, identifying
π1(P1 Δ) with π1(S F∗) = h1,h2,h3, where S ⊂ Φ is the section y = −32, the
generators h1,h2 are as showninFig. 1, and h3 is asimilar loop aboutthe fiberF−1,
notshowninthefigure.(Aslongash1,h2,h3generateπ1(P1 Δ),theparticularchoice
of h3 isirrelevant.Forthereader’sconvenience,wefixittobelcl−1,wherec isthecircle
t → −1+12exp(2πt) about −1 andl isthesemicircle t → 12exp(πt) circumventing the origin.)Thus,onearrivesat thepresentation
π1
Φ (V ∪ F∗)= v1, v2, v3, v4, h1, h2, h3h−1i vjhi= βi(vj)
,
where i = 1,2,3, j = 1,2,3,4, and βi ∈ Autv1,v2,v3,v4 is the so-called braid
mon-odromy,i.e.,theautomorphismofthefundamentalgroupobtainedbydraggingthefiber along hiwhilekeepingthebasepointin S.(Theformaldefinitionisintermsofa trivial-izationoftheinducedfibration(p◦ hi)∗p overthesegment[0,1],wherep◦ hiisregarded as amap [0,1]→ P1 Δ;foralldetails,see[11,16].)
Now, inorder topassto π1(Φ V ),oneneedstopatch intheonlyspecialfiberF−1
fact,theprincipalapplicationofthetheoremin[16]isthefollowingsimpleobservation, whichwestateinaslightlygeneralizedform.
Lemma 3.1. Let X be a smooth quasi-projective surface and D ⊂ X a closed smooth irreduciblecurve.Thentheinclusionhomomorphismπ1(X D)→ π1(X) isan
epimor-phism; its kernel is normally generated by the class [∂Γ ], where Γ is an analytic disc transversalto D atitscenteranddisjointfrom D otherwise. 2
SinceD isassumedirreducible,theconjugacyclassof[∂Γ ] inthestatementdoesnot depend on the choice of Γ or path connecting ∂Γ to the basepoint. The proof of the lemmaisliterally thesameasin[16], usingatubular neighborhoodof D.
Applying Lemma 3.1 to the curve F−1 V in Φ V , we obtain an extra relation
h3 = 1. In other words, we disregard the generator h3 and convert the four relations
h−13 vjh3= β3(vj) intovj= β3(vj),j = 1,2,3,4.
The computation of the braid monodromy is straightforward and well known, e.g.,
usingequationsofthelines;itislefttothereader.(Essentially,itisthebraidmonodromy ofthenodalarrangementL1+L2+R2+R3offourlines.)Denotingbyσ1,σ2,σ3theArtin
generators[2]ofthebraidgroupB4actingonv1,v2,v3,v4,wehaveβ1= σ12σ32,β2= σ22,
and β3 = σ1−1σ3−1σ22σ3σ1. (It is worth recalling that, assuming the left action of the
automorphismgroup,thebraidmonodromyisananti-homomorphismπ1(P1Δ)→ B4.)
Indeed,β1andβ2areessentiallycomputedintheveryfirstpaperonthesubject,viz.[11]:
each is the local monodromy about a simple node (one full twist of a pair of points abouttheirbarycenter) orapair ofdisjoint nodes.Theremaining braid β3 isthelocal
monodromyσ22about−1 translatedvia l (seethedefinitionofh3above)tothecommon
referencefiber;thetranslationconjugatesthelocalmonodromyby‘onehalf’of β1,which
isσ1σ3.
Putting everything together, after a slightsimplification the nontrivial relations for thefundamentalgroupπ1(Φ V ) take theform
[h2, v1] = [h2, v4] = 1, (3.2)
h2v2v3= v2v3h2= v3h2v2 (3.3)
(therelationsh−12 vjh2= β2(vj) fromthefiberx= 1),
h1v1v2= v1v2h1= v2h1v1, (3.4)
h1v3v4= v3v4h1= v4h1v3 (3.5)
(therelationsh−11 vjh1= β1(vj) fromthefiberx= 0), and
v−12 v1v2, v4
= 1 (3.6)
(therelations vj = β3(vj) from the fiber x=−1). For the last relation (3.6), onecan consider a local geometric basis vi := σ−11 σ−13 (vi), i = 1,. . . ,4, over x = −12; inthis
basis, the relation is [v2,v3] = 1 and, on the other hand, one has v2 = v2−1v1v2 and
v3= v4(theresultofcircumventingtheorigin).
ByLemma 3.1,theinclusionin:Φ V → Φ R inducesthemap
in∗: π1↠ G : h1 → g1, v2 → g2, v3 → g3, h2, v1, v4 → 0.
3.2. The‘universal’ covering
Throughout the paper we use freely the following well-known fact, often referred to as theory of covering spaces: for any connected, locally path connected, and micro-simply connectedtopologicalspace X (e.g.,forany connectedsimplicialcomplex)with a basepoint x0 ∈ X, there is anatural equivalence between the category of coverings
( ˜X,x˜0)→ (X,x0) andcoveringmaps(identicalonX)andthatofsubgroupsofπ1(X,x0)
and inclusions. If the subgroup is normal (regular, or Galois coverings), it can be de-scribed as thekernel of anepimorphism α:π1(X,x0)↠ G;theimage G servesthen as
thegroupofthedecktranslationsofthecovering.
Consider an epimorphism α:G ↠ G.In this section, we donot assume G finite; in fact, we startwith astudy of the ‘universal’G-covering,corresponding to theidentity map 0:G ↠ G/0G = G. (Admittedly awkward, this notation is in perfect agreement with m:G↠ G/mG introduced earlier.)
Considerthecomposition
˜ α: π1 in∗ −−↠ π1(Φ R) = G α −−↠ G
and denote byΦ◦[α] the G-coveringof Φ V definedby ˜α.Bythe Hurewicz theorem,
H1(Φ◦[α]) istheabelianizationofπ1(Φ◦[α])= Ker ˜α.Theactionofthedecktranslations
ofthecoveringmakesthisgroupaZ[G]-module;regardedassuch,itisoftenreferredto as theAlexandermodule of ˜α.
The construction of the Alexander module fits into a more general framework and admitsapurelyalgebraicdescription.Consideragroupπ andanepimorphismα:˜ π↠ G
to an abelian group G. Then the Alexander module of ˜α is the abelian group A := Ker ˜α/[Ker ˜α,Ker ˜α] regardedasaZ[G]-modulevia theG-actiondefinedasfollows:given
a∈ A andg∈ G, theimageg(a) is theclassinA oftheelement g˜˜a˜g−1 ∈ Ker ˜α,where ˜
a,g˜∈ π aresomeliftsofa,g,respectively.This class doesnotdepend onthechoiceof thelifts,andtheactioniswell defined.
Crucial is the fact that H1(Φ◦[α]) depends on the epimorphism α:˜ π1 ↠ G only.
Hence, we can replace Φ V with any CW -complex X with π1(X) = π1. We take
for X a space with a single 0-cell e0, one 1-cell e1
i ∈ {a1,a2,a3,c1,c2,c3} for each of
the six generators h1,v2,v3,h2,v4,v1 of π1 (in the order listed), and one 2-cell e2j for eachrelation(3.2)–(3.6).IntheG-covering X[0],eachcell e givesrisetoawholeG-orbit
assumethatthelabelling iscompatiblewith theG-action,i.e.,for anycell e in X and pairh,g∈ G wehaveh(g⊗ e)= (h+ g)⊗ e.)
Followingthetradition,letus identifyZ[G] withthering
Λ :=Zt±11 , t±12 , t±13
ofLaurentpolynomials,wherethevariablest1,t2,t3correspondtothegeneratorsh1 →
g1,v2 → g2,v3 → g3 aboutR1,R2,R3,respectively.Inotherwords,weidentifyG with
themultiplicativeabeliangroupgeneratedbyt1,t2,t3;wewillalsousethismultiplicative
notation inthe celllabels. We canassume, inaddition, that thelabelling ischosen so thattheleftendofeach‘initial’1-cell1⊗ e isattachedto1⊗ e0,i.e.,(1⊗ e)(0)= 1⊗ e0.
(Here,weregardanoriented1-cellasapath[0,1]→ X[0].)Then,fromthedefinitionof thecoveringitfollows thattherightends areattachedasfollows:
(1⊗ ai)(1) = ti⊗ e0, (1⊗ cj)(1) = 1⊗ e0, i, j = 1, 2, 3, (3.7)
i.e., the generators h1,v2,v3 are ‘unwrapped’, whereas h2,v1,v4 remain ‘latent’. The
otherendsaredeterminedbytheG-action:fora1-cell e in X,amonomialt int1,t2,t3,
and= 0,1 wehave(t⊗ e)()= t((1⊗ e)()).
Recallthatthemember CnofthecellularchaincomplexassociatedtoaCW -complex
Y isthe freeabelian groupgenerated bythe n-cellsof Y .Thus, eachcell e of X gives
riseto adirectsummand Z(g ⊗ e), g∈ G, inthecomplexof X[0];this summand is naturallyidentifiedwiththefreeΛ-moduleΛe.(Itisthisidentificationthatexplainsthe usageof⊗ in thelabels.) Furthermore,sincetheCW -structure onX[0] isG-invariant,
theboundaryhomomorphismsareΛ-linear.Thus,thechaincomplexC∗:= C∗[0] ofX[0]
isacomplexoffreeΛ-modulesoftheform 0−→ C2
∂2
−→ C1= Λa1⊕ Λa2⊕ Λa3⊕ Λc1⊕ Λc2⊕ Λc3
∂1
−→ C0= Λ−→ 0
(weomitthegeneratore0 ofC0),where∂1 isgivenby(3.7):
∂1ai= (ti− 1), ∂1cj = 0, i, j = 1, 2, 3. (3.8) Themodule C2 hasninegenerators,ofwhichsixhavenon-trivialimagesunder ∂2:
(t2t3− 1)c1, (3.9)
(t3− 1)c1+ (t3− 1)a2− (t2− 1)a3 (3.10)
from(3.3),
(t1t3− 1)c2, (3.11)
from (3.5),and
(t1t2− 1)c3, (3.13)
(t1− 1)c3+ (t1− 1)a2− (t2− 1)a1 (3.14)
from (3.4).Relations(3.2)and(3.6)contribute 0 to Im ∂2.
Example3.15.Theproofof(3.9)–(3.14)isastraightforwardcomputation.Asanexample, consider (3.3),whichcanbe writtenintheformoftwo relations
h2v2v3h−12 v−13 v−12 = 1, h2v2v3v2−1h−12 v−13 = 1.
The word intheleft handside of thefirst relationcorresponds to the sequence c1,a2,
a3, c−11 , a−13 ,a−12 of1-cells in X alongwhich a2-cell e21 is attached.(Theinverse fora
1-cell means the reversionof theorientation.) Lift this sequence to X[0],starting each cell attheendof thepreviousone,see(3.7):
1⊗ c1, 1⊗ a2, t2⊗ a3, (t2t3⊗ c1)−1, (t2⊗ a3)−1, (1⊗ a2)−1.
(Observe that,forexample,t2⊗ a3 connectst2⊗ e0tot2t3⊗ e0,see(3.7);hence,thelift
of a−13 starting att2t3⊗ e0 is (t2⊗ a3)−1; itends at t2⊗ e0. Notealso that(1⊗ a2)−1
endsat1⊗ e0,i.e.,theliftisaloop,asexpected.)Weobtainasequenceof1-cellsalong which a2-cellinX[0],viz.oneof theliftsof e21, isattached; writingthis sequenceas a chain, weget ∂2e21 = (1− t2t3)c1∈ C1, whichis (3.9)upto sign.Similarly, thesecond
relation liftsto thesequence
1⊗ c1, 1⊗ a2, t2⊗ a3, (t3⊗ a2)−1, (t3⊗ c1)−1, (1⊗ a3)−1,
whichgivesus(3.10).
3.3. Othercoverings
Now,givenanepimorphismα:G↠ G,itinducesaringhomomorphismα∗:Λ↠ Z[G],
makingZ[G] aΛ-module.Clearly,theG-coveringX[α] isthequotientspaceX[0]/Ker α, the cells in X[α] being the Ker α-orbits of those in X[0]. The chain homomorphism
C∗ → C∗(X[α]) induced bythequotientprojection merelyidentifiesthebasiselements (whicharethecells) withineachorbitofKer α;algebraically,itcanbeexpressedas the tensor product
id⊗ α∗: C∗= C∗⊗ΛΛ−→ C∗⊗ΛZ[G] = C∗
X[α].
Recall, see the beginningof Section3.2, thatthe 1-homology of thecoveringspaces dependonlyonthehomomorphismα:˜ π1↠ G.Hence,thegroupH1(Φ◦[α])= H1(X[α])
Coker(∂2⊗Λα∗) = (Coker ∂2)⊗Λ Z[G], our primary interest is the quotient A[α] :=
C1[α]/Im ∂2.Explicitly,A[α] canbedescribedastheΛ-modulegeneratedbythesix
ele-mentsa1,a2,a3,c1,c2,c3 thataresubjecttorelations(3.9)–(3.14)andtheextrarelation
tr1 1 t r2 2 t r3 3 = 1 whenever α(r1g1+ r2g2+ r3g3) = 0. (3.16)
Summarizing,aftertheidentificationC0[α]=Z[G] andH0(X[α])=Z,wehaveanexact
sequence
0−→ H1
Φ◦[α] −→ A[α] ∂1
−→ Z[G] −→ Z −→ 0, (3.17) wherethelasthomomorphismis theaugmentationg → 1,g∈ G.
Recall that the rank rk A of a finitely generated abelian group A is the maximal number of linearly independent elements of A, whereas its length (A) isthe minimal numberofelementsgenerating A.Onehasrk A= (A) ifandonlyifA isfree.
Lemma 3.18. For any epimorphism α:G ↠ G, there is a natural isomorphism
Tors H1(Φ◦[α]) = Tors A[α]. If G is finite, then (H1(Φ◦[α])) = (A[α])− |G|+ 1 and
rk H1(Φ◦[α])= rk A[α]− |G|+ 1.
Proof. Since Im ∂1 ⊂ Z[G] is a free abelian group, the inclusion in (3.17) induces an
isomorphismof the torsion parts. This isomorphismand theobvious fact that(A) = rk A+ (Tors A) for any finitely generated abelian group A imply thatthe length and rankidentitiesinthestatementareequivalent to eachother. Therankidentity follows fromtheadditivityofrankin(3.17)and theobservationthatrkZ[G]=|G|. 2
3.4. Fermatsurfaces
If the image G of α:G ↠ G is finite, one obviously has Φ◦[α] = Φ[α] V [α]. If
α = m ∈ Z, i.e., in thecase of a classical Fermat surfaceΦ[m], it is more convenient to consider asmaller divisorL[m]¯ := L[m]+ R0[m], see (2.4).The fundamentalgroup
π1(Φ[m] ¯L[m]) is givenbyLemma 3.1: itis thequotientof Ker ˜α = π1(Φ◦[α]) by the
extrarelationshm
1 = v2m= v3m= 1 (astheramificationindexateachcomponentofR[m]
is obviously m). Hence, thehomology groupH1(Φ[m] ¯L[m]) canbe computed using
the complex C∗[m] with three extra 2-cells e2
i, i = 1,2,3, mapped by ∂2 to ϕm(ti)ai, where
ϕn(t) :=
tn− 1/(t− 1), n ∈ Z.
This computation is similar to Example 3.15: for example, the loop hm
1 lifts to the
sequence 1⊗ a1,t1⊗ a1,t21 ⊗ a1,. . . ,tm−11 ⊗ a1 of 1-cells, which results in the chain
(1+ t1+ t21+ . . . + tm1−1)a1= ϕm(t1)a1∈ C1[m].Note thatthischainis acycle,as in
Remark3.19. Strictlyspeaking,thenew complexisthatof abeliangroupsrather than
Λ-modules, as we add three 2-cells only,i.e., three summands Ze2i in C2[m]. However,
in the presence of therelations tm
i = 1, i = 1,2,3, cf. (3.16), one canuse (3.9)–(3.14) to show that all three images ϕm(ti)ai are G-invariant. Hence, without changing the 1-homology of thecomplex, we canformally replace eachsummand Ze2
i with Λe2i, ex-tending ∂2 by Λ-linearity. Geometrically, we replace a single disk Γ as in Lemma 3.1
with aG-orbit consisting of m3 disks. Since the curve Ri[m] patched in is irreducible (alldisksintersectingthesamecomponent),thischangedoesnotaffectthefundamental group.
Now, asinSection3.3,insteadofextendingtheC2-termofthecomplex,wecanadd
extrarelationsto C1.Summarizing,wehave
H1
Φ[m] ¯L[m]= Ker∂1: ¯A[m]→ C0[m]
,
where A[m] is¯ thequotientofA[0] bytheextrarelations
tmi = 1, ϕm(ti)ai= 0, i = 1, 2, 3. (3.20) Arguingas intheproofof Lemma 3.18,weobtaintheidentity
H1
Φ[m] ¯L[m]= A[m]¯ − m3+ 1. (3.21)
3.5. OtherDelsarte surfaces
Inthegeneralizedcase,thefirstquestionthatarisesiswhetherTheorem 2.2is appli-cable,i.e.,whetherH1(Φ[α])= 0.To statetheresult,introducethefollowingnotation:
given apair of integers0 i,j 3, let Gij :=Zgi⊕ Zgj ⊂ G, where gi ∈ G are the canonicalgenerators,seeSection2.4.
Recall that the blow-up σ:X˜ → X of a smooth point of a surface X induces an isomorphism of both the fundamental group π1 and first homology group H1 of the
surface. Hence,up to canonicalisomorphism, thegroupsπ1 and H1 donot depend on
theresolution ofsingularities.
Proposition 3.22.Foran epimorphismα:G↠ G,|G|<∞,onehas π1
Φ[α]= H1
Φ[α]= Ker α/Gij∩ Ker α,
thesummation runningover allpairs0 i,j 3 ofintegers.
Proof. We startwiththeabeliangroupπ1(Φ R)=G generatedbyh1 → g1,v2 → g2,
v3 → g3,seeSection3.1.Clearly,π1(Φ[α]R[α])= H1(Φ[α]R[α])= Ker α.(Thisgroup
Fortherestoftheproof,weusetheadditivenotationforthefundamentalgroup(asthe groupsofinterestaresubquotientsofG).
LetΦ[α] bethemanifoldobtainedfrom Φ[α] R[α] bypatching thecomponentsof thepropertransformofR[α] awayfromtheexceptionaldivisor.Atagenericpointof Ri, the ramification index mi of the ramified covering Φ[α] → Φ equals the index [Gii : Gii∩ Ker α], i= 0,1,2,3.Hence,byLemma 3.1, theinclusioninducesanepimorphism Ker α ↠ π1(Φ[α]) whose kernel is generated by the elements migi. Thus, we have an isomorphism π1 Φ[α]= Ker α/ i Gii∩ Ker α, i = 0, 1, 2, 3. (3.23) (Strictly speaking, unlike the case of the Fermat surfaces, the curve Ri[α] may be re-ducible,sothatweneedtoattachaseparatedisk Γ asinLemma 3.1foreachcomponent ofthiscurve.However,sincetheG-actionistrivialinthe1-homologyH1= π1,alldisks
resultinthesamerelationmigi= 0,cf.Remark 3.19.)
Whatremains ispatching theexceptional divisors.Fixapair 0 i< j 3 andlet ˜
S be asingularpoint of thenormalized, butyet unresolved ramified coveringover the pointS := Ri∩ Rj.Fixaresolutionofsingularitiesandlet E betheexceptionaldivisor over ˜S. Picka sufficiently small ball U ⊂ Φ about S and denote by ˜U the connected componentofthepreimage of U containing E.Withrespect toanappropriate smooth triangulation,U is˜ aregularneighborhoodof E;hence,E isastrictdeformationretract of ˜U ,U˜∼ E.Ontheotherhand,U is˜ a4-manifoldwithboundary∂ ˜U ,andthelatteris acoveringofthe3-sphere∂U ramifiedover theHopflinkR∩ ∂U.
Notealsothatthecontractionof E givesusthespaceU /E which˜ istheconeover ∂ ˜U
(withthevertex S = E/E);˜ hence,wehaveahomotopyequivalence(strictdeformation retraction)U˜ E = ( ˜U /E) ˜S ∼ ∂ ˜U .
Wehaveπ1(∂UR)=Gijand,hence,π1(∂ ˜UR[α])=Gij∩Ker α.Asabove,similar to Lemma 3.1, patching theunion of circles ∂ ˜U ∩ R[α] results in the pair of relations
migi= mjgj= 0.Thus,
π1(∂ ˜U ) = (Gij∩ Ker α)/(Gii∩ Ker α + Gjj∩ Ker α) (3.24) isafinitegroup.ThenH1(∂ ˜U ;Q)= 0,i.e.,∂ ˜U isarationalhomologysphereandS is˜ a
rationalsingularpoint.Forus,importantisthefactthatπ1( ˜U )= π1(E)= 0,whichcan
easilybe proveddirectly.Indeed,sinceU˜ ∼ E anddimRE = 2,we haveH3( ˜U ;Q)= 0; thenalsoH1( ˜U ,∂ ˜U ;Q)= 0 (Lefschetzduality),andthefragment
H1(∂ ˜U ;Q) −→ H1( ˜U ;Q) −→ H1( ˜U , ∂ ˜U ;Q)
ofthe homologyexactsequence ofpair ( ˜U ,∂ ˜U ) impliesH1( ˜U ;Q)= H1(E;Q)= 0. On
theotherhand,E isaconnectedprojectivealgebraic curve,anditiseasilyseenthatE
thenormalizationof E)andanumberofcircles.(Roughly,wecan‘blow-up’thelocally reducible singular points of E to line segments, separating the analytic branches and replacing E with a disjoint union of topologically nonsingular closed surfaces with a numberofsegmentsattached.Then,withineachsurface,movetheendsofthesegments toasinglepoint.Finally,contractseveralsegmentstomakethesurfacesshareacommon point; the result is a wedge as stated.) For such a wedge E ∼ iEi, all groups are easily computed (e.g., using iteratedly the Mayer–Vietoris exact sequence (8.8) in [7, Chapter III]andSeifert–van Kampentheorem[16],or justdecomposingthewedgeinto cells):
H1(E;Q) =
i
H1(Ei;Q), π1(E) =∗iπ1(Ei).
Clearly, H1(E;Q)= 0 ifand onlyifallsurfacecomponentsare2-spheresand there are
nocirclespresent. Thenobviouslyπ1(E)= 0.
Now, startwith Φ[α] andproceedpatching theexceptionaldivisorsonebyone.Let
Φ be an intermediate space, notyet containing E.Applying the Seifert–van Kampen theorem[16]totheunionΦ∪ ˜U andusingthehomotopyequivalenceΦ∩ ˜U = ˜UE ∼ ∂ ˜U ,weobtaintheamalgamatedfreeproduct
π1 Φ∪ ˜U=π1 Φ∗ π1( ˜U ) /π1(∂ ˜U ) = π1 Φ/(Gij∩ Ker α).
(Forthelastisomorphism,weuse(3.24)andtheidentityπ1( ˜U )= 0.)Thegroupπ1(Φ[α])
isgivenby(3.23)and, afteralltheexceptionaldivisorshavebeenpatched, wearriveat theexpression inthestatement. 2
IfH1(Φ[α])= 0,Theorem 2.2andLemma 3.18implythat
T V [α] ∼= Tors A[α]. (3.25) Unfortunately, as aZ[G]-module, A[α] is far from freeand it is difficult to controlits Z-torsion. (Experimentsshowthat,atleast,theintermediatequotients similartothose consideredinLemma 4.4dooftenhavetorsion.)Anattemptofadirectcomputationis madein Section4.3,whereasinthecaseoftheclassicalFermatsurfaceswehavetotake adetourand estimatethelengthinstead.Thefollowingtwoexactsequencesmayprove useful:
A[α] ∂1
−→ Z[G]
−→ Z −→ 0,
where istheaugmentation,see(3.17),and
where A◦[α] ⊂ A[α] is the submodule generated by c1, c2, c3. The former sequence
merelystatesthatH0(C∗[α])= H0(Φ◦[α])=Z.Forthelatter, wepatchL[α] (byusing
Lemma 3.1ormerelyforgettingthegeneratorsh2,v1,v4,hencec1,c2,c3inthefirstplace)
tocomputethegroupH1(Φ[α] R[α])= π1(Φ[α] R[α])= Ker α;theresultingcomplex
is0→ A[α]/A◦[α]→ Z[G]→ 0.Both sequencessplit,and wecanextend(3.25)to T V [α] ∼= Tors A◦[α] = Tors A[α], (3.26) stillundertheassumption thatH1(Φ[α])= 0.
4. Proofofthemaintheorem
4.1. The lengthofA[m]¯
Fix an integer m 1 and consider the Λ-module A[m] introduced¯ in Section 3.4. RecallthatA[m] is¯ generatedbysixelementsai,cj,i,j = 1,2,3,subjecttotherelations (3.9)–(3.14)and(3.20).Observethatrelations(3.9), (3.11),and(3.13) canbe recastin theform
tick= t−1j ck whenever{i, j, k} = {1, 2, 3}. (4.1) Weintroduceafewad hoc notations.Giveni= 1,2,3,let
Λi:=Z[ti]/
tmi − 1, Λ¯i:=Z[ti]/ϕm(ti).
These rings can be regarded as Λ-modules, but we usually do not specify the action of the other two variables: it varies from case to case. In fact, we repeatedly use the followingsimpleobservation,whichisanimmediateconsequenceof(4.1).
Lemma4.2. Let i,j,k∈ {1,2,3},k= i, and p∈ Λ,and letA be asubquotient of A[m]¯ generatedbyasingleelementx:= pci.Assumethateithertj = 1 orti= t±1k on A.Then
A is aquotientofΛsx foranappropriate indexs∈ {1,2,3}.
Ifx is alsoannihilatedby ϕm(ts),then A is aquotientofΛ¯sx. 2
Theprecise descriptionofthe‘appropriate’index s (notnecessarilyunique)isleftto thereader.Clearly,(Λs)= m and( ¯Λs)= m− 1.
Forageneratorx∈ {a1,a2,a3,c1,c2,c3},let
x:= (t1− 1)x, x := (t˜ 3− 1)x, x˜ := (t1− 1)˜x.
Observethatalways
We will useafiltration0= A0 ⊂ A1 ⊂ . . . ⊂ A7= ¯A[m], whereAk ⊂ ¯A[m] are the submodules definedinLemma 4.4below.
Letδm:= 1 ifm isevenandδm:= 0 ifm isodd.
Lemma 4.4.One has thefollowingequationsand inequalities:
(1) (A1/A0)= m3− m2,whereA1 is thesubmodule generatedby a3;
(2) (A2/A1) 3(m− 1)− δm,where A2:= A1+ Λ˜a2+ Λ˜c3;
(3) (A3/A2) 3(m− 1), whereA3:= A1+ (t3− 1) ¯A[m];
(4) (A4/A3)= m2− m,whereA4:= A3+ Λa1;
(5) (A5/A4) m− 1, whereA5:= A4+ Λa2+ Λc3;
(6) (A6/A5)= m− 1, whereA6:= A5+ Λa2;
(7) (A7/A6) 2m+ 1,whereA7:= ¯A[m].
Hence, (A) m3+ 9m− 7− δ
m.
Proof. One has (A1) m2(m− 1) due to (3.20). On the other hand, the boundary
homomorphism ∂1 maps A1 onto (t3− 1)C0[m]. Hence, there are no other relations
in A1,andstatement (1)holds. Furthermore,∂1factorsto ahomomorphism
¯
A[m]/A3→ C0 := C0[m]/(t3− 1)
which maps A4/A3 isomorphically onto (t1− 1)C0, proving statement (4). Then, ∂1
factorsto
¯
A[m]/A5→ C0:= C0/(t1− 1) = Λ2.
Since A6/A5 is(apriori aquotientof) thecyclicΛ¯2-moduleΛ¯2a2,therestrictionof ∂1
maps itisomorphicallyonto(t2− 1)C0= ¯Λ2, provingstatement (6).
Fortheotherstatements,itsufficestoestimatethenumberofgenerators.With possi-blefutureapplicationsinmind,wedescribethestructureoftheintermediatequotientsin theform(known module)↠ Ak/Ak−1.Infact,alltheseepimorphismsareisomorphisms, see Remark 4.14below.
InA[m]/A¯ 4,onehas
t3= 1, a1= a3= 0, a2=−c3;
the last relation follows from (3.14). Thus, A5/A4 is generated by c3, and A[m]/A¯ 6 is
generated byc1,c2,c3;by(3.20)andLemma 4.2,
¯
ForthelastsummandZc3,weusethefactthat
(t1− 1)c3=−(t1− 1)a2= 0 mod A6.
Thus,( ¯A[m]/A6) 2m+ 1,andstatements (5)and (7)areproved.
ThemoduleA3/A1isgeneratedbya˜1,˜a2,c˜1,˜c2,˜c3,andrelations(3.10),(3.12),(3.14)
imply
˜
a2=−˜c1, ˜a1=−˜c2, (t1− 1)(˜c3+ ˜a2) = (t2− 1)˜a1.
Wecanretain threegenerators˜c1,˜c2,c˜3 only,rewriting thelast relationintheform
(t1− 1)(˜c3− ˜c1) + (t2− 1)˜c2= 0. (4.6)
Notealsothatϕm(t3)A3= 0,see (4.3).
InA3/A2, wehave(t1− 1)˜c3= (t1− 1)˜a2= 0,hencealso(t1− 1)˜c1 = 0.Then(4.6)
implies(t2− 1)˜c2= 0,and
¯
Λ3˜c1⊕ ¯Λ3c˜2⊕ ¯Λ3c˜3↠ A3/A2, (4.7)
seeLemma 4.2.Thisgivesusstatement (3).
ThemoduleA2/A1 isgenerated by ˜c1 and ˜c3.By(4.3)and(4.1),wehave
ϕm(ti)(A2/A1) = 0 for all i = 1, 2, 3. (4.8)
Relations(3.11)and(4.6)imply(t1t3−1)(˜c3− ˜c1)= 0;using(4.1),thiscanberewritten
as(t3− t2)˜c3= (t1− t2)˜c1.Let
u := (t3− t2)˜c3= (t1− t2)˜c1
andconsider thecyclic submoduleA2⊂ A2/A1 generatedby u.ByLemma 4.2,
¯
Λ2˜c1⊕ ¯Λ2˜c3↠ (A2/A1)/A2. (4.9)
On the other hand, A2 ⊂ Λ˜c1∩ Λ˜c3; hence, t−13 = t2 = t−11 on this module and, by
Lemma 4.2again,
¯
Λ2u↠ A2 if m is odd. (4.10)
Thisfactprovesstatement (2)inthecaseofm odd.
If m = 2k is even, (4.10) still holds, but we need a stronger statement. Note that
ϕm(t) isdivisiblebyϕk(t2).Furthermore,onehasapolynomialidentity
tm−2 m−1 r=0 t1−rϕr t2= tϕk−1t2ϕm(t) + ϕk t2, (4.11)
which is easily establishedby multiplying both sides by t2− 1. On the submoduleA2
we havet2 = t−11 , see (4.1); hence, s := t2t−11 = t22. Then, representing u in the form
u= t1(1− s)˜c1,wehave t12−rϕr t22u = t12−rt1ϕr(s)(1− s)˜c1= tr1 1− sr˜c1=t1r− tr2c˜1, r∈ Z. (4.12) Summingupoverr = 0,. . . ,m− 1 andusing(4.8)and(4.11)att= t2,weconcludethat
ϕk(t22)u= 0,i.e.,
Λ2u/ϕk
t22↠ A2 if m = 2k is even, (4.13) obtainingastrongerinequality(A2) deg ϕk(t2)= m− 2.
Thefinal inequalityinthestatementofthelemmais thesumofitems (1)–(7). 2
4.2. Proofof Theorem 1.3
Weassumethatm 3.By(2.4),itsufficestoshowthatT¯L[m]= 0,whereL[m]¯ :=
L[m]+ R0[m] isthe divisorintroduced inSection3.4. Since Φ[m] is simplyconnected,
we can use Theorem 2.2, reducing the problem in question to proving the inequality
(H1(Φ[m] ¯L[m])) rk K¯L[m].
Accordingto[1,14],rk Sm= 3(m− 1)(m− 2)+ 1+ δm.Ontheotherhand,H2( ¯L[m])
is the free abelian groupgenerated by the classes of the 3m2 lines and the additional
class [R0[m]].Hence,rk K¯L[m]= 9m− 6− δm,andthestatementfollows from(3.21) and Lemma 4.4. 2
Remark4.14.ItfollowsfromtheproofthatallinequalitiesinthestatementofLemma 4.4 are, in fact,equalities, i.e., norelation hasbeen lost, even thoughsomerelations were multiplied bynon-units.Furthermore,allepimorphisms (4.5),(4.7),(4.9),(4.10),(4.13) are isomorphisms.
Remark4.15.Weonlyusetheinequalityrk Sm 3(m− 1)(m− 2)+ 1+ δm,i.e.,thefact thatthereisatleast a certainnumberofrelationsbetweenthecomponents.Ingeneral, itwouldsufficetoprovetheinequality(A[α]) rk KV [α]+|G|− 1,seeLemma 3.18. Remark4.16.Therankrk Sm caneasilybecomputeddirectly,bytensoringthemodule by C and counting the irreducible summands, which are all of dimension 1 (multi-eigenspaces ofthethreecommutingfiniteorder operatorst1,t2,t3).
Remark 4.17. By (3.26), when computing the torsion, one can replace A[α] with the smaller module A◦[α]. A posteriori, A◦[m] is the Λ[m]-module spanned by the three generatorsc1,c2,c3 subjecttoasinglerelation
see [5].In this form,someof theresults ofthis paper generalize to Fermatvarieties of higherdimension,see [6].Note, though,thatthisone-relatorpresentationofA◦[α] does notextendto moregeneralDelsartesurfaces;see[5]forfurtherdetails.
4.3. A toyexample
Inconclusion,weconsideraverysimpleexample,answeringthegeneralizedquestion, seeProblem 2.6,inthespecialcaseofacoveringramifiedover atmostthreelines. Theorem4.18. Ifthecovering Φ[α]→ Φ is unramifiedoveratleast oneof thelines Rj,
j = 0,1,2,3, thenTV [α]= 0.
Proof. We can assume that the covering is unramified over R3, i.e., the epimorphism
α:G↠ G sends g3 to 0.Then,obviously, Ker α =Zg3⊕ (G12∩ Ker α) and,by
Proposi-tion 3.22,we haveH1(Φ[α])= 0,i.e.,Theorem 2.2isapplicable.
By(3.16),wehavet3= 1 onA[α],and relations(3.10),(3.12), (3.14)become
(t2− 1)a3= (t1− 1)a3= 0, (t1− 1)(c3+ a2) = (t2− 1)a1.
Introducingthegeneratora2:= c3+ a2insteadof a2,weseethatthesubmoduleA◦[α]⊂
A[α] introduced inSection3.5 is adirect summand (as aΛ-module), and all relations inA◦[α] aret3= 1 and (3.9),(3.11),(3.13).Thethreelattertranslateinto independent
relations(t2− 1)c1 = (t1− 1)c2 = (t1t2− 1)c3 = 0, and A◦[α] isadirectsum ofthree
grouprings: A◦[α] =ZG/α(g2) c1⊕ Z G/α(g1) c2⊕ Z G/α(g1+ g2) c3.
By(3.26),onehasTV [α]∼= Tors A◦[α]= 0. 2
Corollary 4.19 (of (2.5) and Theorem 4.18). If a covering pr:Φ[α] → Φ as in Theo-rem 4.18has degree m primeto 6, thenPic Φ[α]= SV [α]. 2
4.4. Proofof Theorem 1.4 andCorollary 1.5
Corollary 1.5isanimmediateconsequenceofTheorem 1.4 andthefactthatPic Σp,q is rationally generated by the classesof the lines, see [3]. In viewof Theorem 2.2, the statementofTheorem 1.4 ispurelyhomological,andwecandeformΣp,q totheFermat surface Φ[m]; then, the m2 lines in question deformto the components of L
1[m], and
Sp,q = SL1[m]. Similar to (2.4), the latter group equals S¯L1[m], where L¯1[m] :=
L1[m]+ R0[m].
PatchingL2[m] andL3[m],cf. Section3.4,weconcludethat
T L¯1[m]
Filtering this module as inLemma 4.4 and analyzing the proof of the lemma, we see thatstatements (1),(4),and (6)holdwithoutchange,whereastheotherstatementscan be rewrittenasfollows:
(2) (A2/A1)= 0 dueto(4.6),
(3) (A3/A2) m− 1,see(4.7),
(5) (A5/A4)= 0,see (4.5),
(7) (A7/A6) m,see(4.5).
Summing this up, we obtain ( ¯A[m]) m3+ 2m− 2. On the other hand, one has
rk S¯L1[m]= (m− 1)2+ 1,see[3];hence,rk K¯L1[m]= 2m− 1 and,asinSection4.2,
we concludethatA¯[m] isafreeabeliangroup. 2 Acknowledgments
I would like to express my gratitude to I. Shimada for bringing the problem to my attention and formany fruitful discussions; itwas hewho eventuallypersuaded me to publishtheseobservations.Iwouldalsoliketothanktheanonymousrefereeofthispaper fortheelegant proofof(4.12).
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