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Desingularization of arithmetic surfaces: algorithmic aspects AnneFru¨hbis-Kru¨gerandStefanWewers 7 1 0 2 n a J 5 AbstractThequestforregularmodelsofarithmeticsurfacesallowsdifferentview- ] pointsandapproaches:usingvaluationsoracoveringbycharts.Inthisarticle,we G sketchbothapproachesandthenshowinaconcreteexample,howsurprisinglyben- A eficial it can be to exploit properties and techniques from both worlds simultane- . ously. h t a m [ 1 Introduction 1 v Resolution of singularities in dimension 2 was first proved by Jung in 1908 [16], 0 butit wasnotuntilHironaka’sworkin 1964[15] thatthis couldalso be mastered 9 in dimensionsbeyond3. However,Hironaka’sresult onlyapplies to characteristic 3 1 zero,butnotto positiveormixedcharacteristic.Therethe generalquestionisstill 0 wide open with partial results for low dimensions. In particular, Lipman gave a . constructionfor2-dimensionalschemesinfullgeneralityin[17]. 1 0 Lipman’sresult includes the case of an arithmetic surface, i.e. integralmodels 7 ofcurvesovernumberfields.Infact,theexistenceof(minimal)regularmodelsof 1 curvesovernumberfieldsisacornerstoneofmodernarithmeticgeometry.Important : v earlyresultsareforinstancetheexistenceofaminimalregularmodelofanelliptic i curvebyNe´ron([23])andTate’salgorithm([30])forcomputingitexplicitly. X In this paper we study a particular series of examples of surface singularites r a which is a specialcase of a constructiondueto Lorenzini([19], [20]). The singularity in question is a wild quotientsingularity. More precisely, the singular point liesonanarithmeticsurfaceofmixedcharacteristic(0,p)whichisthequotientof aregularsurfacebyacyclicgroupofprimeorder p,suchthatthegroupactionhas Institutfu¨rAlgebraischeGeometrie,LeibnizUniversita¨tHannover e-mail:[email protected] Institutfu¨rReineMathematik,Universita¨tUlm e-mail:[email protected] 1 2 Fru¨hbis-Kru¨ger,Wewers isolatedfixedpoints.We provethatinourexampleoneobtainsaseriesofrational determinantalsingularitiesofmultiplicity p,andweareabletowritedownexplicit equationsforthese(seeProposition3.4). Determinantalrings(ofexpectedcodimension)arewell-studiedobjectsincom- mutative algebra: the free resolution is the Eagon-Northcott complex and hence many invariants of the ring such as projective dimension, depth, Castelnuovo- Mumford regularity, etc. are known (see e.g. [8], [3]). Beyond that, such singularities(in thegeometriccase)are an activeareaof currentresearchin singularity theorystudyinge.g.classificationquestions,invariants,notionsofequivalenceand topologicalproperties,see e.g.[10],[24],[32].We show,bya directcomputation, thattheresolutioninourarithmeticsettingiscompletelyanalogoustothegeometric case. Both for derivingthe equationsof our singularitiesand for resolving them, we employandmix two ratherdifferentapproachesto representandto computewith arithmeticsurfaces.Thefirstapproachismorestandardandconsistsinrepresenting a surfaces as a finite union of affine charts, and the coordinate ring of each affine chartasafinitelygeneratedalgebraoverthegroundring.Fromthispointofview, computationswith arithmetic surfaces can be performedwith standard tools from computeralgebra,likestandardbases(e.g.inSINGULAR[6]).However,thesetech- niquesarenotyetasmatureinthearithmeticcaseastheyareinthegeometriccase. The secondapproachuses valuationsas its main tool. We work overa discrete valuationringR.AnarithmeticsurfaceXoverSpecRisconsideredasanR-modelof itsgenericfiberX (asmoothcurveoverK=Frac(R)).Thenany(normal)R-model K X of X is determined by a finite setV(X) of discrete valuations on the function K field of X correspondingto the irreduciblecomponentsof the special fiber of X. K A priori,it is notclear how to extractusefulinformationaboutthe modelX from thesetV(X).Nevertheless,injointworkwithJ.Ru¨ththesecondnamedauthorhas usedthistechniquesuccessfullyforcomputingsemistablereductionofcurves(see e.g.[28]). Thepaperisstructuredasfollows.InSection2wegivesomegeneraldefinitions concerningarithmeticsurfaces,andwepresentourtwoapproachesforrepresenting themexplicitly.Section3thenpresentsourseriesofwildquotientsingularities.In thefinalsection,wecompute,inoneconcreteexampleofourwildquotientsingu- larities,anexplicitdesingularization. 2 Arithmetic surfaces andmodels ofcurves 2.1 General definitions Definition2.1.By a surface we mean an integraland noetherianscheme X of di- mension2.AnarithmeticsurfaceisasurfaceX togetherwithafaithfullyflatmor- phism f :X →S=Spec(R)offinitetype,whereRisaDedekinddomain.Toavoid Desingularizationofarithmeticsurfaces:algorithmicaspects 3 technicalities, we always assume that R (and hence X) is excellent. Moreover,we willassumeinadditionthatX isnormal,unlessweexplictlysayotherwise. Acommonsituationwherearithmeticsurfacesoccuristhefollowing.LetRbea Dedekinddomain,K=Frac(R)andX asmoothandprojectivecurveoverK.An K R-modelofX isanarithmeticsurfaceX→Spec(R),togetherwithanidentification K ofX withthegenericfiberofX,i.e.X =X⊗ K. K K R ForthefollowingdiscussionwefixanarithmeticsurfaceX→Spec(R).Wewrite Xsing forthesubsetofpointswhoselocalringisnotregular.Sinceweassumethat X is normal,Xsing is closed of codimension2 and henceconsists of a finite set of closed pointsof X. A pointx ∈Xsing is called a singularity of X. (If we dropthe normalitycondition,thenXsingmayalsohavecomponentsofcodimension1.) ByamodificationofX wemeanaproperbirationalmap f :X′→X.Amodifi- cationisanisomorphismoutsideafinitesetofclosedpoints.If f isanisomorphism awayfromasinglepointx ∈X,thenx iscalledthecenterofthemodificationand E := f−1(x )⊂X′ theexceptionalfiberorexceptionallocus(weendowE withthe reducedsubschemestructure).NotethatEisaconnectedschemeofdimensionone. Wewillusethenotation E=∪n C, i=1 i wheretheC aretheirreduciblecomponents.Eachofthemisaprojectivecurveover i theresiduefieldk=k(x ).Ifthemodificationchangesmorethanasinglepoint,we will still denote the exceptional locus by E, but E obviously does not need to be connectedanymore. Definition2.2.Let p:X →S beanarithmeticsurfaceandx ∈Xsing asingularity. A desingularizationof x ∈X is a modification f :X′ →X with center x and ex- ceptionalfiberE = f−1(x )suchthateverypointx ′∈E isaregularpointofX′.A desingularizationofX isamodificationconsistingofdesingularizationsatallpoints ofXsing. By a theorem Lipman ([17]), a desingularization of X always exists by means ofasequenceofnormalizationsandblow-ups.Dependingonthesituationweoften want f tosatisfyfurtherconditions.Welistsomeofthem: (a) TheexceptionaldivisorE isanormalcrossingdivisorofX′. (b) Lets:= p(x).ThenthefiberX′ ofX′ oversisanormalcrossingdivisoronX′ s (whenendowedwiththereducedsubschemestructure). (c) The desingularization f :X′ →X is minimal(amongall desingularizationsof x ∈X). (d) f :X′→X isminimalamongalldesingularizationssatisfying(a)(resp.(b)). Choosing a differentapproach than Lipman and avoiding normalizationscom- pletely,Cossart,JanssenandSaitoprovedadesingularizationalgorithmrelyingonly onblow-upsatregularcentersin[4],seealso[5].Theapproachallowstoaddition- allysatisfyyetanotherrathercommoncondition: 4 Fru¨hbis-Kru¨ger,Wewers (e) IfX⊂W forsomeregularscheme1,thendesingularizationofXcanbeachieved bymodificationsofW whichareisomorphismsoutsideXsing. 2.2 Presentationbyaffine charts We are interested in the problem of computing a desingularization f :X′ →X of a given singularity x ∈X on an arithmetic surface explicitly. Before we can even statethisproblemprecisely,wehavetosaysomethingaboutthewayinwhichthe surfaceX isrepresented. Themostobviousway2topresentX istowriteitasaunionofaffinecharts, X =∪r U , U =SpecA . j=1 j j j Here each A is a finitely generatedR-algebrawhose fractionfield is the function j fieldF(X)ofX.AfterchoosingasetofgeneratorsofA /R,wecanobtainapresen- j tation‘bygeneratorsandrelations’.Thismeansthat A =R[x]/I , j j wherex=(x ,...,x )isasetofindeterminatesandI ✁R[x]isanideal.Choosing 1 nj j alistofgeneratorsofI ,weobtainapresentation j R[x]mj →R[x]→A →0. j TakingintoaccounttherelationsamongthegeneratorsoftheidealI thispresenta- j tionextendsto R[x]nj →R[x]mj →R[x]→A →0, j where the matrix describing the left-most map is usually referred to as the first syzygy matrix of I or A respectively. Iteratively forming higher syzygies, this j j leads to free resolutions, i.e. exact sequences of free R[x]-modules. As R[x] is a polynomialringoveraDedekinddomain,ithasglobaldimensionn +1andhence j A possesses a freeresolutionof lengthatmostn +1. Workinglocallyat a max- j j imal ideal m⊂R[x], this allows e.g. the calculation of the m-depth of A by the j Auslander-Buchsbaumformula. Inthesubsequentsections,weshallencounterexamplesplacingusinaparticular situation, for which free resolutions are well understood: determinantal varieties correspondingtomaximalminors.Forthese,I isgeneratedbythemaximalminors j ofanm×nmatrixdefiningavarietyofcodimension(m−t+1)(n−t+1),where t=min{m,n}.Mostprominently,theHilbert-Burchtheorem(seeforinstance[8]) relatesCohen-Macaulaycodimension2varietiestothet-minorsoftheirfirstsyzygy 1asbeforeW shouldbeexcellent,noetherian,integral 2thankstoGrothendieck Desingularizationofarithmeticsurfaces:algorithmicaspects 5 matrix,whichisofsizet×(t+1),andensuresthemapgivenbythismatrixtobe injective. 2.3 Presentationusing valuations Analternativeway3 topresentanarithmeticsurfaceisthefollowing.Todescribeit itisconvenienttoassumethatRisalocalring.ThenRisactuallythevaluationring ofadiscretevaluationv :K×→QofitsfractionfieldK=Frac(R).Wechoosea K uniformizerp ofv (i.e.ageneratorofthemaximalidealp✁R)andnormalizev K K such thatv (p )=1. We denotethe residuefield of v byk. In additionwe make K K thefollowingassumption4: Assumption2.3.Thevaluationv iseitherhenselian,oritsresiduefieldk isalge- K braicoverafinitefield. WefixasmoothprojectivecurveX overK.NotethatX isuniquelydetermined K K byitsfunctionfieldF ,andconverselyeveryfinitelygeneratedfieldextensionF/K X oftranscendencedegree1isthefunctionfieldofasmoothprojectivecurveX . K LetX beanR-modelofX ,X itsspecialfiberand K s X =∪X¯ s i i itsdecompositionintoirreduciblecomponents.TheneachcomponentX¯ isaprime i divisoronthesurfaceX.BecauseX isnormal,X¯ givesrisetoadiscretevaluation i v on F such that v(p )>0. We normalize v such that v(p )=1. i.e. such that i X i i i v| = v . By definition, the residue field k(v) of v is the function field of the i K K i i componentX¯.Inparticular,k(v)isfunctionfieldoverkoftranscendencedegree1. i i AdiscretevaluationvonthefunctionfieldF iscalledgeometricifv| =v and x K K theresiduefieldk(v)isafinitelygeneratedextensionofkoftranscendencedegree 1. LetV(F ) denote the set of geometric valuations. Given a model X of X , we X K write V(X):={v ,...,v }⊂V(F ) 1 r X forthesetofgeometricvaluationscorrespondingtothecomponentsofthespecial fiberofX. Theorem2.4.Themap X 7→V(X) isabijectionbetweenthesetofisomorphismclassesofR-modelsofX andtheset K offinitenonemptysubsetsofV(F ). X 3 Historically, thiswas actuallythe first method, pioneered by Deuring [7] more than 10 years beforetheinventionofschemes. 4Moregenerally,wecouldhaveassumedthat(K,vK)satisfiesthelocalSkolemproperty,see[13] 6 Fru¨hbis-Kru¨ger,Wewers Furthermore,giventwomodelsX,X′ ofX ,thereexistsamapX′→X whichis K the identity on X (and which is then automatically a modification) if and only if K V(X)⊂V(X′). Proof. See[12]or[27]. ⊓⊔ BytheabovetheoremmodelsofagivensmoothprojectivecurveX overaval- K uedfield(K,v )canbedefinedsimplybyspecifyingafinitelistofvaluations.An K obviousdrawbackofthisapproachisthatitisnotobvioushowto extractdetailed information on the model X from the setV(X). A priori,V(X) only gives ‘birational’informationonthe specialfiberX . Forinstance,it is notimmediateto see s whetherthemodelX isregular. Sofar,theaboveapproachbasedonvaluationshasprovedtobeveryusefulfor the computation of semistable models (see [28]). We intend to extend it to other problemsinthefuture.In§4.2wewillseeafirstattempttouseitfordesingulariza- tion. 2.4 Computationaltools Inthissectionwereportonsomeongoingworktoimplementcomputationaltools fordealingwitharithmeticsurfacesandtheirdesingularization. Valuationbasedapproach Aswehaveexplainedin§2.3,itisinprinciplepossibletodescribearithmeticsur- facesoveralocalfield purelyin termsofvaluations.Inordertousethisapproach forexplicitcomputations,oneneedsawaytowritedown,manipulateandcompute withgeometricvaluations.Fortunately,suchmethodsareavailable(butmaybenot as widely known as they should). Our approach goes back to work of MacLane ([21], [22]). In the presentcontext(i.e. for describingmodelsof curvesoverlocal fields)ithasbeendevelopedsystematicallyinJulianRu¨th’sPhDthesis([27]). We willnotgointodetails,butforlateruse weneedtointroducethe notionof an inductivevaluation.LetK be a fieldwith a discretevaluationv andvaluation K ringRasbefore.Letvbeanextensionofv toageometricvaluationontherational K functionfieldK(x).Weassumeinadditionthatv(x)≥0(i.e.thatR[x]iscontained in the valuation ring of v). Let f ∈R[x] be a monic integral polynomial, and let l ∈Qbearationalnumbersatifyingl >v(f ).Iff isakeypolynomialforv(see [27], Definition 4.7) then we can define a new geometric valuation v′ (called an augmentationofv)withthepropertythat v′(f )=l , v′(f)=v(f) for f ∈K[x]withdeg(f)<deg(f ). See[27],Definition4.9.Wewrite Desingularizationofarithmeticsurfaces:algorithmicaspects 7 v′=[v, v′(f )=l ]. The process of augmenting a given geometric valuation can be iterated. A geometric valuation v on K(x) which is obtained by a sequence of augmentations, startingfromtheGaussvaluationwithrespecttox,iscalledaninductivevaluation. Itcanbewrittenas v=v =[v ,v (f )=l ,...,v (f )=l ]. (1) n 0 1 1 1 n n n Herev istheGaussvaluation,l ∈Qandf ∈R[x]ismonic.Furthermore,f isa 0 i i i keypolynomialforv andl >v (f ).By[27],Theorem4.31,everygeometric i−1 i i−1 i valuationvonK(x)withv(x)≥0canbewrittenasaninductivevaluation. The notion of inductive valuation can be extended in several ways. Firstly, by replacing x with x−1 if necessary, we can drop the condition v(x)≥0, Hence we can write every geometric valuation on K(x) as an inductive valuation. Secondly, forthelastaugmentationstepin(1) wecanallowthevaluel =¥ . Theresulting n v isthenonlyapseudo-valuationandinducesatruevaluationonthequotientring n L:=K[x]/(f )(whichisafieldbecausekeypolynomialsareirreducible).Thirdly, n givenanarbitraryfiniteextensionL/K,wecancomputethe(finite)setofextensions wofv toLasfollows.We writeL=K[x]/(f)foranirreduciblepolynomial f ∈ K K[x]. If f is irreducibleoverthe completionKˆ of K with respectto v , then there K existsauniqueextensionwofvtoLwhichcanbewrittenasaninductivepseudo- valuationon K[x] (with f = f). In general, let f =(cid:213) f be the factorizationinto n i i irreducibles over Kˆ. Then each factor f gives rise to an extension w of v to L. i i Consideringw asapseudo-valuationonK[x],MacLaneshowsthatw canbewritten i i asalimitvaluationofachainofinductivevaluationsv .Bythiswemeanthatv is n n anaugmentationofv ,andforeverya =(g(x) mod(f))∈Lthereexistsn≥0 n−1 suchthatw(a )=v (g)=v (g)=.... i n n+1 MacLane’stheoryis constructiveand can be used to implementalgorithmsfor dealingwithdiscretevaluationsonafairlylargeclassoffields.ASagepackagewrit- tenbyJulianRu¨thcalledmac lane([26])isavailabelundergithub.com/saraedum/mac_lane. Itcanbeusetodefineandcomputewithdiscretevaluationsofthefollowingkind: • p-adicvaluationsonnumberfields. • GeometricvaluationsvonfunctionfieldsF/K(ofdimension1)whoserestriction toK iseithertrivial,orcanbedefinedbythispackage. GivenavaluationvonafieldK oftheabovekindandafiniteseparableextension L/K,itispossibletocomputethesetofallextensionofvtoK. Chartbasedapproach Ontheotherhand,adescriptionbyaffinechartsasin2.2notonlyemphasizesthe similaritytothegeometricsetting,italsoallowstheuseofcomputationaltechniques suchasstandardbases(wheneverasuitablypowerfularithmeticforcomputationsin Risavailable).This,inturn,opensupawholeportfolioofalgorithmsrangingfrom 8 Fru¨hbis-Kru¨ger,Wewers basic functionality like elimination or ideal quotients to more sophisticated algo- rithmssuchasblowingupandnormalization,whicheventuallypermittopractically implementtheabovementionedalgorithmsofLipmanandofCossart-Janssen-Saito fordesingularizationof2-dimensionalschemes.Noteatthispointthatneitherofthe twoalgorithmsimposestheconditionofnormalityonthesurfacestoberesolved. In a nutshell, the desingularization problem for 2-dimensional schemes is the problemoffindingsuitablecenterswhichimprovethesingularitywithoutintroduc- ingnewcomplications.Inthiscontext,0-dimensionalcentersforblow-upsusually do not pose any major problems:such blow-upsat differentcentersmay be inter- changed,astheyareisomorphismsoutsidetheirrespectivecentersandhencedonot interact. However, even resolvinga 0-dimensionalsingular pointin the geometric casemayalreadyrequiretheuseof1-dimensionalcenterstoachievearegularmodel andnormalcrossingdivisors.Thesecurvescanexhibitsignificantlymorestructure than sets of points, e.g. they can possess intersecting components or non-regular branches. So the central problems in resolving the singularities of 2-dimensional schemes are ensuring improvement in each step and treating 1-dimensional loci whichneedtobeimproved.Inparticularforthelatter,thetwoaforementionedap- proachesdiffersignificantly. ThekeyideabehindLipman’salgorithm[17]isthatnormalvarietiesareregular incodimension1,i.e.thattheirsingularlocusis0-dimensional.Thusanormaliza- tionstepcanalwaysensurethatonlysetsofpointswillberequiredforsubsequent blowingup: Theorem2.5([17]). LetX be an excellent, noetherian,reducedscheme ofdimension 2, then X posses a desingularization by a finite sequence of birational mor- phismsoftheform X p−r→◦nr···p−2→◦n2X p−1→◦n1X =X, r 1 0 wherep denotesablowupatafinitenumberofpoints,n anormalizationandX i i r isregular. While blowing up is algorithmically straightforward e.g. using an elimination (seee.g.[9]),thehardstepisthenormalization.Althoughtherehasbeensignificant improvementin the efficiencyofGrauert-Remmertstyle normalizationalgorithms inthelastdecade(seee.g.[14],[1]),thisisstillabottleneckwhenworkingovera DedekinddomainRinsteadofafield.Thecrucialstephereisthechoiceofasuit- abletestideal,i.e.aradicalidealcontainedintheidealofthenon-normallocusand containinganon-zerodivisor.Inthegeometriccase,theidealofthesingularlocus– generatedbytheoriginalsetofgeneratorsandtheappropriateminorsoftheJaco- bianmatrix–iswell-suitedforthistask,butinthecurrentsettingitalsoseesfibre singularitieswhich do not contributeto the non-regularlocus. Hence the approxi- mationofthe non-normallocusby thistest idealis rathercoarseandsignificantly impedes efficiency.In practice, a better approximationof the non-normallocus is achieved by constructing a test ideal following an idea of Hironaka’s termination Desingularizationofarithmeticsurfaces:algorithmicaspects 9 criterion:weusethelocuswhereHironaka’sinvariantn ∗,i.e.thetupleoforders(in thesenseofordersofpowerseries)oftheelementsofalocalstandardbasis,sorted byincreasingorder,islexicographicallygreaterthanatupleofones. The approach of Cossart-Janssen-Saito [4] (CJS for short) on the other hand, avoids normalization completely and allows well-chosen 1-dimensional centers, whenevernecessary;whenchoosingcenters,ittakesintoaccountthefullhistoryof blowingupsleadingtothecurrentsituation.InconstrasttoLipman’sapproach,this algorithmyieldsan embeddeddesingularization.Nevertheless,a key step is again theuseofthelocuswheren ∗lexicographicallyexceedsatupleofones.Butthen,no normalizationfollows,insteadthesingularitiesofthislocusarefirstresolvedbefore itisitselfusedasa1-dimensionalcenter.Eacharisingexceptionalcurveinthispro- cessrememberswhenitwascreatedandwhetheritscenterwasofdimension0or1, becausethisinformationiscrucialinthechoiceofcenterforensuringimprovement aswellasnormalcrossingofexceptionalcurves. A beta version of the first algorithm is available as SINGULAR-library reslip- man.libandisplannedtobecomepartofthedistributioninthenearfuture.Apro- totype implementation of the CJS-algorithm has been implemented and is closely relatedtoanongoingPhD-projectonaparallelapproachtoresolutionofsingular- itiesusingthegpi-spaceparallelizationenvironment(forrecentprogressalongthis trainofthoughtsee[2],[25]). 3 Explicitconstruction ofwildquotient singularities Inthissectionwedescribeaseriesofexamplesforarithmeticsurfaceswithinteres- tingsingularities.ThegeneralconstructionisduetoLorenzini(see[19]and[20]). Ourcontributionistoexplictlydescribethe(local)ringofthesingularitybygener- atorsandrelations.Inthenextsectionwealsodescribethedesingularizationinan equallyexplicitway. LetR be a discrete valuationring, with maximalidealp, residue field k=R/p andfractionfield K. Letv denotethe correspondingdiscretevaluationonK. We K assume that k has positive characteristic p and that v is henselian (in particular, K Assumption2.3holds). LetX beasmooth,projectiveandabsolutelyirreduciblecurveoverK,ofgenus K g.We assumethatX haspotentiallygoodreductionreductionwithrespecttov . K K ThismeansthatthereexistsafiniteextensionL/K andasmoothmodelY ofX := L X ⊗ LovertheintegralclosureR ofRinL.NotethatR isadiscretevaluation K K L L ring correspondingto the unique extension v of v to L. We assume in addition L K thatL/K isaGaloisextension,andthatthenaturalactionofG:=Gal(L/K)onX L extendstoanactiononY.Underthisassumption,wecanformthequotientscheme X /G.ItisanR-modelofX . Y K ThemodelY isregularbecauseY→Spec(R)issmoothbyassumption.However, thequotientschemeX =Y/Gmayhavesingularities.Infact,letx ∈X beaclosed s 10 Fru¨hbis-Kru¨ger,Wewers pointonthespecialfiberofX,andleth ∈Ysbeapointabovex .LetIh ⊂Gdenote theinertiasubgroupofh inG.IfIh =1thenthemapY →X ise´taleinh .Itfollows thatX isregularinx becauseY isregularinh . In general, the locus of points with Ih 6=1 may consists of the entire closed fiberY and hence be a subset of codimension1 onY. To obtain isolated quotient s singularitiesweimposethefollowingcondition: Assumption3.1.TheactionofGonthespecialfiberY isgenericallyfree. s Under this assumption, there are at most a finite numberof points h ∈Y with s nontrivialinertiaIh 6=1.Letx 1,...,x r∈Xs betheimagesofthepointsh ∈Ys with Ih 6=1.Thenx 1,...,x r arepreciselythesingularitiesofthemodelX. Remark3.2.InLorenzini’soriginalsetting,Assumption3.1holdsautomaticallybe- causethecurveY hasgenusg(Y)≥2.Inourseriesofexampleswehaveg(Y)=0, buttheassumptionholdsnevertheless. 3.1 An explicitexample Let p be a primenumber,K a numberfield andp| p a prime idealof O over p. K Let v denote the discrete valuationon K correspondingto p and R the valuation K ringofv . LetL/K bea Galoisextensionof degree p whichis totallyramifiedat K p. This meansthatv has a uniqueextensionv to L. Lets be a generatorof the K L cyclicgroupG=Gal(L/K).Letp beauniformizerforv .Wenormalizev such L L L thatv (p )=1.Set L L m:=v (s (p )−p ). L L L Then m≥2 is the first and only breakin the filtraton of G by higher ramification groups.Weletu∈k× denotetheimageoftheelement(s (p )−p )/p m∈R×. L l L LetX :=P1 betheprojectivelineoverK.WeidentifythefunctionfieldF with K K X therationalfunctionfieldK(x)intheindeterminatex.ThenL(x)isthefunctionfield ofX =P1.Wedefineanelement L L x−p L y:= ∈L(x). p m L Clearly, L(x)=L(y), and so y, considered as a rational functionon X , gives rise L toanisomorphismXL∼=P1L.WeletY denotethesmoothRL-modelofXL suchthat y extends to an isomophismY ∼=P1 . By an easy calculation we see that s (y)= RL ay+b,witha∈R×andb∈R .Furthermore, L L s (y)≡y+u (mod p ). L In geometric terms this means that the action of G on X extends to the smooth L modelY,andthattherestrictionofthisactiontothespecialfiberYs∼=P1k isgenericallyfree(andhenceAssumption3.1holds).Infact,theactionofGisfixpointfree