Desingularization of arithmetic surfaces: algorithmic aspects

0.17 MB·

by  Anne Frühbis-Krüger

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 singu-
larity in question is a wild quotientsingularity. More precisely, the singular point
liesonanarithmeticsurfaceofmixedcharacteristic(0,p)whichisthequotientof
aregularsurfacebyacyclicgroupofprimeorder p,suchthatthegroupactionhas
Institutfu¨rAlgebraischeGeometrie,LeibnizUniversita¨tHannover
e-mail:[e-mail]
Institutfu¨rReineMathematik,Universita¨tUlm
e-mail:[e-mail]
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 singu-
larities(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 ‘bira-
tional’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 ge-
ometric 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 ofdimen-
sion 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 isgeneri-
callyfree(andhenceAssumption3.1holds).Infact,theactionofGisfixpointfree