BERTINI THEOREMS OVER FINITE FIELDS BJORN POONEN Abstract - PDF document

BERTINITHEOREMSOVERFINITEFIELDSBJORNPOONENAbstract.LetXbeasmoothquasiprojectivesubschemeofPnofdimensionm0overFq.ThenthereexisthomogeneouspolynomialsfoverFqforwhichtheintersectionofXandthehypersurfacef=0issmooth.Infact,thesetofsuchfhasapositivedensity,equaltoX(m+1)�1,whereX(s)=ZX(q�s)isthezetafunctionofX.AnanalogueforregularquasiprojectiveschemesoverZisproved,assumingtheabcconjectureandanotherconjecture.1.IntroductionTheclassicalBertinitheoremssaythatifasubschemeXPnhasacertainproperty,thenforasucientlygeneralhyperplaneHPn,H\Xhasthepropertytoo.Forinstance,ifXisaquasiprojectivesubschemeofPnthatissmoothofdimensionm0overaeldk,andUisthesetofpointsuinthedualprojectivespacePncorrespondingtohyperplanesHPn(u)suchthatH\Xissmoothofdimensionm�1overtheresidueeld(u)ofu,thenUcontainsadenseopensubsetofPn.Ifkisinnite,thenU\Pn(k)isnonempty,andhenceonecanndHoverk.Butifkisnite,thenitcanhappenthatthenitelymanyhyperplanesHoverkallfailtogiveasmoothintersectionH\X.SeeTheorem3.1.N.M.Katz[Kat99]askedwhethertheBertinitheoremoverniteeldscanbesalvagedbyallowinghypersurfacesofunboundeddegreeinplaceofhyperplanes.(Infactheaskedforalittlemore;seeSection3fordetails.)Weanswerthequestionarmativelybelow.O.Gab-ber[Gab01,Corollary1.6]hasindependentlyprovedtheexistenceofgoodhypersurfacesofanysucientlylargedegreedivisiblebythecharacteristicofk.LetFqbeaniteeldofq=paelements.LetS=Fq[x0;:::;xn]bethehomogeneouscoordinateringofPn,letSdSbetheFq-subspaceofhomogeneouspolynomialsofdegreed,andletShomog=S1d=0Sd.Foreachf2Sd,letHfbethesubschemeProj(S=(f))Pn.Typically(butnotalways),Hfisahypersurfaceofdimensionn�1denedbytheequationf=0.DenethedensityofasubsetPShomogby(P):=limd!1#(P\Sd) #Sd;ifthelimitexists.ForaschemeXofnitetypeoverFq,denethezetafunction[Wei49]X(s)=ZX(q�s):=YclosedP2X�1�q�sdegP
�1=exp 1Xr=1#X(Fqr) rq�rs!: Date:August31,2004.1991MathematicsSubjectClassication.Primary14J70;Secondary11M38,11M41,14G40,14N05.ThisresearchwassupportedbyNSFgrantDMS-9801104andaPackardFellowship.PartoftheresearchwasdonewhiletheauthorwasenjoyingthehospitalityoftheUniversitedeParis-Sud.ThisarticlehasbeenpublishedinAnnalsofMath.160(2004),no.3,1099{1127.1 2BJORNPOONENTheorem1.1(Bertinioverniteelds).LetXbeasmoothquasiprojectivesubschemeofPnofdimensionm0overFq.DeneP:=ff2Shomog:Hf\Xissmoothofdimensionm�1g:Then(P)=X(m+1)�1.Remarks.(1)Theemptyschemeissmoothofanydimension,including�1.Later(forinstance,inTheorem1.3),wewillsimilarlyusetheconventionthatifPisapointnotonaschemeX,thenforanyr,theschemeXisautomaticallysmoothofdimensionratP.(2)Inthispaper,\denotesscheme-theoreticintersection(whenappliedtoschemes).(3)Ifn2,thedensityisunchangedifweinsistalsothatHfbeageometricallyintegralhypersurfaceofdimensionn�1.ThisfollowsfromtheeasyProposition2.7.(4)Thecasen=1,X=A1,isawellknownpolynomialanalogueofthefactthatthesetofsquarefreeintegershasdensity(2)�1=6=2.SeeSection5foraconjecturalcommongeneralization.(5)ThedensityisindependentofthechoiceofembeddingX,!Pn!(6)By[Dwo60],Xisarationalfunctionofq�s,soX(m+1)�12Q.Theoverallplanoftheproofistostartwithallhomogeneouspolynomialsofdegreed,andthenforeachclosedpointP2XtosieveoutthepolynomialsfforwhichHf\XissingularatP.TheconditionthatPbesingularonHf\Xamountstom+1linearconditionsontheTaylorcoecientsofadehomogenizationoffatP,andtheselinearconditionsareovertheresidueeldofP.ThereforeoneexpectsthattheprobabilitythatHf\XisnonsingularatPwillbe1�q�(m+1)degP.AssumingthattheseconditionsatdierentPareindependent,theprobabilitythatHf\XisnonsingulareverywhereshouldbeYclosedP2X�1�q�(m+1)degP
=X(m+1)�1:Unfortunately,theindependenceassumptionandtheindividualsingularityprobabilityesti-matesbreakdownoncedegPbecomeslargerelativetod.Thereforewemustapproximateouranswerbytruncatingtheproductafternitelymanyterms,saythosecorrespondingtoPofdegreer.Themaindicultyoftheproof,aswithmanysieveproofs,isinbound-ingtheerroroftheapproximation,i.e.,inshowingthatwhendr1,thenumberofpolynomialsofdegreedsievedoutbyconditionsattheinnitelymanyPofdegreerisnegligible.InfactwewillproveTheorem1.1asaspecialcaseofthefollowing,whichismoreversatileinapplications.TheeectofTbelowistoprescribetherstfewtermsoftheTaylorexpansionsofthedehomogenizationsoffatnitelymanyclosedpoints.Theorem1.2(BertiniwithTaylorconditions).LetXbeaquasiprojectivesubschemeofPnoverFq.LetZbeanitesubschemeofPn,andassumethatU:=X�(Z\X)issmoothofdimensionm0.FixasubsetTH0(Z;OZ).Givenf2Sd,letfjZbetheelementofH0(Z;OZ)thatoneachconnectedcomponentZiequalstherestrictionofx�djftoZi,wherej=j(i)isthesmallestj2f0;1:::;ngsuchthatthecoordinatexjisinvertibleonZi.DeneP:=ff2Shomog:Hf\Uissmoothofdimensionm�1,andfjZ2Tg: BERTINITHEOREMSOVERFINITEFIELDS3Then(P)=#T #H0(Z;OZ)U(m+1)�1:UsingaformalismanalogoustothatofLemma20of[PS99],wecandeducethefollowingevenstrongerversion,whichallowsustoimposeinnitelymanylocalconditions,providedthattheconditionsatmostpointsarenomorestringentthantheconditionthatthehyper-surfaceintersectagivennitesetofvarietiessmoothly.Theorem1.3(Innitelymanylocalconditions).ForeachclosedpointPofPnoverFq,letPbenormalizedHaarmeasureonthecompletedlocalring^OPasanadditivecompactgroup,andletUPbeasubsetof^OPwhoseboundary@UPhasmeasurezero.AlsoforeachP,xanonvanishingcoordinatexj,andforf2SdletfjPbetheimageofx�djfin^OP.AssumethatthereexistsmoothquasiprojectivesubschemesX1;:::;XuofPnofdimensionsmi=dimXioverFqsuchthatforallbutnitelymanyP,UPcontainsfjPwheneverf2ShomogissuchthatHf\Xiissmoothofdimensionmi�1atPforalli.DeneP:=ff2Shomog:fjP2UPforallclosedpointsP2Png:Then(P)=YclosedP2PnP(UP).Remark.ImplicitinTheorem1.3istheclaimthattheproductQPP(UP)alwaysconverges,andinparticularthatitsvalueiszeroifandonlyifP(UP)=0forsomeclosedpointP.TheproofsofTheorems1.1,1.2,and1.3arecontainedinSection2.ButthereaderatthispointisencouragedtojumptoSection3forapplications,andtoglanceatSection5,whichshowsthattheabcconjectureandanotherconjectureimplyanaloguesofourmaintheoremsforregularquasiprojectiveschemesoverSpecZ.Theabcconjectureisneededtoapplyamultivariablegeneralization[Poo03]ofA.Granville'sresult[Gra98]aboutsquarefreevaluesofpolynomials.Forsomeopenquestions,seeSections4and5.7,andalsoConjecture5.2.TheauthorhopesthatthetechniqueofSection2willproveusefulinremovingthecondi-tion\assumethatthegroundeldkisinnite"fromothertheoremsintheliterature.2.Bertinioverfinitefields:theclosedpointsieveSections2.1,2.2,and2.3aredevotedtotheproofsofLemmas2.2,2.4,and2.6,whicharethemainresultsneededinSection2.4toproveTheorems1.1,1.2,and1.3.2.1.Singularpointsoflowdegree.LetA=Fq[x1;:::;xn]betheringofregularfunc-tionsonthesubsetAn:=fx06=0gPn,andidentifySdwiththesetofdehomogenizationsAd=ff2A:degfdg,wheredegfdenotestotaldegree.Lemma2.1.IfYisanitesubschemeofPnoveraeldk,thenthemapd:Sd=H0(Pn;OPn(d))!H0(Y;OY(d))issurjectiveforddimH0(Y;OY)�1.Proof.LetIYbetheidealsheafofYPn.Thencoker(d)iscontainedinH1(Pn;IY(d)),whichvanishesford1byTheoremIII.5.2bof[Har77].Thusdissurjectiveford1.EnlargingFqifnecessary,wecanperformalinearchangeofvariabletoassumeYAn:=fx06=0g.Dehomogenizebysettingx0=1,sothatdisidentiedwithamapfromAd 4BJORNPOONENtoB:=H0(Y;OY).Letb=dimB.Fori�1,letBibetheimageofAiinB.Then0=B�1B0B1:::,soBj=Bj+1forsomej2[�1;b�1].ThenBj+2=Bj+1+nXi=1xiBj+1=Bj+nXi=1xiBj=Bj+1:SimilarlyBj=Bj+1=Bj+2=:::,andtheseeventuallyequalBbythepreviousparagraph.Hencedissurjectivefordj,andinparticularfordb�1.IfUisaschemeofnitetypeoverFq,letUbethesetofclosedpointsofUofdegreer.SimilarlydeneU&#x-278;r.Lemma2.2(Singularitiesoflowdegree).LetnotationandhypothesesbeasinTheorem1.2,anddenePr:=ff2Shomog:Hf\Uissmoothofdimensionm�1atallP2U,andfjZ2Tg:Then(Pr)=#T #H0(Z;OZ)YP2U�1�q�(m+1)degP
:Proof.LetU=fP1;:::;Psg.LetmibetheidealsheafofPionU,letYibetheclosedsubschemeofUcorrespondingtotheidealsheafm2iOU,andletY=SYi.ThenHf\UissingularatPi(moreprecisely,notsmoothofdimensionm�1atPi)ifandonlyiftherestrictionofftoasectionofOYi(d)iszero.HencePr\SdistheinverseimageofTsYi=1�H0(Yi;OYi)�f0g
undertheFq-linearcompositiond:Sd=H0(Pn;OPn(d))!H0(Y[Z;OY[Z(d))'H0(Z;OZ)sYi=1H0(Yi;OYi);wherethelastisomorphismisthe(noncanonical)untwisting,componentbycomponent,bydivisionbythed-thpowersofvariouscoordinates,asinthedenitionoffjZ.ApplyingLemma2.1toY[Zshowsthatdissurjectiveford1,so(Pr)=limd!1#[TQsi=1(H0(Yi;OYi)�f0g)] #[H0(Z;OZ)Qsi=1H0(Yi;OYi)]=#T #H0(Z;OZ)sYi=1�1�q�(m+1)degPi
;sinceH0(Yi;OYi)hasatwo-stepltrationwhosequotientsOU;Pi=mU;PiandmU;Pi=m2U;Piarevectorspacesofdimensions1andmrespectivelyovertheresidueeldofPi.2.2.Singularpointsofmediumdegree.Lemma2.3.LetUbeasmoothquasiprojectivesubschemeofPnofdimensionm0overFq.IfP2Uisaclosedpointofdegreee,whereed=(m+1),thenthefractionoff2SdsuchthatHf\Uisnotsmoothofdimensionm�1atPequalsq�(m+1)e.Proof.LetmbetheidealsheafofPonU,andletYbetheclosedsubschemeofUcorre-spondingtom2.Thef2Sdtobecountedarethoseinthekernelofd:H0(Pn;O(d))!H0(Y;OY(d)).WehavedimH0(Y;OY(d))=dimH0(Y;OY)=(m+1)ed,sodissurjectivebyLemma2.1,andtheFq-codimensionofkerdequals(m+1)e. BERTINITHEOREMSOVERFINITEFIELDS5Denetheupperandlowerdensities (P), (P)ofasubsetPSas(P)wasdened,butusinglimsupandliminfinplaceoflim.Lemma2.4(Singularitiesofmediumdegree).LetUbeasmoothquasiprojectivesubschemeofPnofdimensionm0overFq.DeneQmediumr:=[d0ff2Sd:thereexistsP2UwithrdegPd m+1suchthatHf\Uisnotsmoothofdimensionm�1atPg:Thenlimr!1 (Qmediumr)=0.Proof.UsingLemma2.3andthecrudebound#U(Fqe)cqemforsomec�0dependingonlyonU[LW54],weobtain#(Qmediumr\Sd) #Sdbd=(m+1)cXe=r(numberofpointsofdegreeeinU)q�(m+1)ebd=(m+1)cXe=r#U(Fqe)q�(m+1)e1Xe=rcqemq�(m+1)e;=cq�r 1�q�1:Hence (Qmediumr)cq�r=(1�q�1),whichtendstozeroasr!1.2.3.Singularpointsofhighdegree.Lemma2.5.LetPbeaclosedpointofdegreeeinAnoverFq.Thenthefractionoff2AdthatvanishatPisatmostq�min(d+1;e).Proof.LetevP:Ad!Fqebetheevaluation-at-Pmap.TheproofofLemma2.1showsthatdimFqevP(Ad)strictlyincreaseswithduntilitreachese,sodimFqevP(Ad)min(d+1;e).Equivalently,thecodimensionofker(evP)inAdisatleastmin(d+1;e).Lemma2.6(Singularitiesofhighdegree).LetUbeasmoothquasiprojectivesubschemeofPnofdimensionm0overFq.DeneQhigh:=[d0ff2Sd:9P2U�d=(m+1)suchthatHf\Uisnotsmoothofdimensionm�1atPg:Then (Qhigh)=0.Proof.IfthelemmaholdsforUandforV,itholdsforU[V,sowemayassumeUAnisane.Givenaclosedpointu2U,chooseasystemoflocalparameterst1;:::;tn2AatuonAnsuchthattm+1=tm+2=


=tn=0denesUlocallyatu.Thendt1;:::;dtnareaOAn;u-basisforthestalk 1An=Fq;u.Let@1;:::;@nbethedualbasisofthestalkTAn=Fq;uofthetangentsheaf.Chooses2Awiths(u)6=0tocleardenominatorssothatDi:=s@igivesaglobalderivationA!Afori=1;:::;n.ThenthereisaneighborhoodNuofuin 6BJORNPOONENAnsuchthatNu\ftm+1=tm+2=


=tn=0g=Nu\U, 1Nu=Fq=ni=1ONudti,ands2O(Nu).WemaycoverUwithnitelymanyNu,sobytherstsentenceofthisproof,wemayreducetothecasewhereUNuforasingleu.Forf2Ad,Hf\Ufailstobesmoothofdimensionm�1atapointP2Uifandonlyiff(P)=(D1f)(P)=


=(Dmf)(P)=0.Nowforthetrick.Let=maxi(degti), =b(d�)=pc,and=bd=pc.Iff02Ad,g12A ,...,gm2A ,andh2Aareselecteduniformlyandindependentlyatrandom,thenthedistributionoff:=f0+gp1t1+


+gpmtm+hpisuniformoverAd.WewillboundtheprobabilitythatanfconstructedinthiswayhasapointP2U�d=(m+1)wheref(P)=(D1f)(P)=


=(Dmf)(P)=0.Bywritingfinthisway,wepartiallydecoupletheDiffromeachother:Dif=(Dif0)+gpisfori=1;:::;m.Wewillselectf0;g1;:::;gm;honeatatime.For0im,deneWi:=U\fD1f=


=Dif=0g:Claim1:For0im�1,conditionedonachoiceoff0;g1;:::;giforwhichdim(Wi)m�i,theprobabilitythatdim(Wi+1)m�i�1is1�o(1)asd!1.(Thefunctionofdrepresentedbytheo(1)dependsonUandtheDi.)ProofofClaim1:LetV1,...,V`bethe(m�i)-dimensionalFq-irreduciblecomponentsof(Wi)red.ByBezout'stheorem[Ful84,p.10],`(deg U)(degD1f):::(degDif)=O(di)asd!1,where UistheprojectiveclosureofU.SincedimVk1,thereexistsacoordinatexjdependingonksuchthattheprojectionxj(Vk)hasdimension1.WeneedtoboundthesetGbadk:=fgi+12A :Di+1f=(Di+1f0)+gpi+1svanishesidenticallyonVkg:Ifg;g02Gbadk,thenbytakingthedierenceandmultiplyingbys�1,weseethatg�g0vanishesonVk.HenceifGbadkisnonempty,itisacosetofthesubspaceoffunctionsinA vanishingonVk.Thecodimensionofthatsubspace,orequivalentlythedimensionoftheimageofA intheregularfunctionsonVk,exceeds +1,sinceanonzeropolynomialinxjalonedoesnotvanishonVk.ThustheprobabilitythatDi+1fvanishesonsomeVkisatmost`q� �1=O(diq�(d�)=p)=o(1)asd!1.ThisprovesClaim1.Claim2:Conditionedonachoiceoff0;g1;:::;gmforwhichWmisnite,Prob(Hf\Wm\U�d=(m+1)=;)=1�o(1)asd!1.ProofofClaim2:TheBezouttheoremargumentintheproofofClaim1showsthat#Wm=O(dm).ForagivenpointP2Wm,thesetHbadofh2AforwhichHfpassesthroughPiseither;oracosetofker(evP:A!(P)),where(P)istheresidueeldofP.IfmoreoverdegP�d=(m+1),thenLemma2.5implies#Hbad=#Aq�where=min(+1;d=(m+1)).HenceProb(Hf\Wm\U�d=(m+1)6=;)#Wmq�=O(dmq�)=o(1)asd!1,sinceeventuallygrowslinearlyind.ThisprovesClaim2.EndofproofofLemma2.6:Choosef2Sduniformlyatrandom.Claims1and2showthatwithprobabilityQm�1i=0(1�o(1))
(1�o(1))=1�o(1)asd!1,dimWi=m�i BERTINITHEOREMSOVERFINITEFIELDS7fori=0;1;:::;mandHf\Wm\U�d=(m+1)=;.ButHf\WmisthesubvarietyofUcutoutbytheequationsf(P)=(D1f)(P)=


=(Dmf)(P)=0,soHf\Wm\U�d=(m+1)isexactlythesetofpointsofHf\Uofdegree�d=(m+1)whereHf\Uisnotsmoothofdimensionm�1.2.4.Proofsoftheoremsoverniteelds.ProofofTheorem1.2.AsmentionedintheproofofLemma2.4,thenumberofclosedpointsofdegreerinUisO(qrm);thisguaranteesthattheproductdeningU(s)�1convergesats=m+1.ByLemma2.2,limr!1(Pr)=#T #H0(Z;OZ)U(m+1)�1:Ontheotherhand,thedenitionsimplyPPrP[Qmediumr[Qhigh,so (P)and (P)eachdierfrom(Pr)byatmost (Qmediumr)+ (Qhigh).ApplyingLemmas2.4and2.6andlettingrtendto1,weobtain(P)=limr!1(Pr)=#T #H0(Z;OZ)U(m+1)�1:ProofofTheorem1.1.TakeZ=;andT=f0ginTheorem1.2.ProofofTheorem1.3.TheexistenceofX1;:::;XuandLemmas2.4and2.6letusapproxi-matePbythesetPrdenedonlybytheconditionsatclosedpointsPofdegreelessthanr,forlarger.ForeachP2Pn,thehypothesisP(@UP)=0letsusapproximateUPbyaunionofcosetsofanidealIPofniteindexin^OP.(ThedetailsarecompletelyanalogoustothoseintheproofofLemma20of[PS99].)Finally,Lemma2.1impliesthatford1,theimagesoff2SdinQP2Pn^OP=IPareequidistributed.FinallyletusshowthatthedensitiesinourtheoremsdonotchangeifinthedenitionofdensityweconsideronlyfforwhichHfisgeometricallyintegral,atleastforn2.Proposition2.7.Supposen2.LetRbethesetoff2ShomogforwhichHffailstobeageometricallyintegralhypersurfaceofdimensionn�1.Then(R)=0.Proof.WehaveR=R1[R2whereR1isthesetoff2ShomogthatfactornontriviallyoverFq,andR2isthesetoff2ShomogoftheformNFqe=Fq(g)forsomehomogeneouspolynomialg2Fqe[x0;:::;xn]ande2.(Note:ifourbaseeldwereanarbitraryperfecteld,anirreduciblepolynomialthatisnotabsolutelyirreduciblewouldbeaconstanttimesanorm,buttheconstantisunnecessaryhere,sinceNFqe=Fq:Fqe!Fqissurjective.)Wehave#(R1\Sd) #Sd1 #Sdbd=2cXi=1(#Si)(#Sd�i)=bd=2cXi=1q�Ni;whereNi=n+dn�n+in�n+d�in: 8BJORNPOONENFor1id=2�1,Ni+1�Ni=n+d�in�n+d�i�1n�n+i+1n�n+in=n+d�i�1n�1�n+in�1�0:Similarly,fordn,N1=n+d�1n�1�n+1nn+d�11�n+11=d�2:Thus#(R1\Sd) #Sdbd=2cXi=1q�Nibd=2cXi=1q2�ddq2�d;whichtendstozeroasd!1.Thenumberoff2Sdthatarenormsofhomogeneouspolynomialsofdegreed=eoverFqeisatmost(qe)(d=e+nn).Therefore#(R2\Sd) #SdXej�d;e1q�MewhereMe=�d+nn
�e�d=e+nn
.For2ed,e�d=e+nn
�d+nn
=e�d e+n
�d e+n�1



�d e+1
(d+n)(d+n�1)


(d+1)e�d e+n
�d e+n�1
(d+n)(d+n�1)e�d e+n
2 d2=1 e+2n d+en2 d21 2+2n2 d+dn2 d22=3;onced18n2.Henceinthiscase,Me1 3�d+nn
d2=6forlarged,so#(R2\Sd) #SdXej�d;e1q�Medq�d2=6;whichtendstozeroasd!1.AnotherproofofProposition2.7isgiveninSection3.2,butthatproofisvalidonlyforn3. BERTINITHEOREMSOVERFINITEFIELDS93.Applications3.1.CounterexamplestoBertini.Ironically,wecanuseourhypersurfaceBertinithe-oremtoconstructcounterexamplestotheoriginalhyperplaneBertinitheorem!Moregen-erally,wecanshowthathypersurfacesofboundeddegreedonotsucetoyieldasmoothintersection.Theorem3.1(Anti-Bertinitheorem).GivenaniteeldFqandintegersn2,d1,thereexistsasmoothprojectivegeometricallyintegralhypersurfaceXinPnoverFqsuchthatforeachf2S1[


[Sd,Hf\Xfailstobesmoothofdimensionn�2.Proof.LetH(1),...,H(`)bealistoftheHfarisingfromf2S1[


[Sd.Fori=1;:::;`inturn,chooseaclosedpointPi2H(i)distinctfromPjforji.UsingaTasinTheorem1.2,wecanexpresstheconditionthatahypersurfaceinPnbesmoothofdimensionn�1atPiandhavetangentspaceatPiequaltothatofH(i)wheneverthelatterissmoothofdimensionn�1atPi.Theorem1.2(withProposition2.7)impliesthatthereexistsasmoothprojectivegeometricallyintegralhypersurfaceXPnsatisfyingtheseconditions.Thenforeachi,X\H(i)failstobesmoothofdimensionn�2atPi.Remark.Katz[Kat99,p.621]remarksthatifXisthehypersurfacen+1Xi=1(XiYqi�XqiYi)=0inP2n+1overFqwithhomogeneouscoordinatesX1;:::;Xn+1;Y1;:::;Yn+1,thenH\XissingularforeveryhyperplaneHinP2n+1overFq.3.2.Singularitiesofpositivedimension.LetXbeasmoothquasiprojectivesubschemeofPnofdimensionm0overFq.Givenf2Shomog,let(Hf\X)singbetheclosedsubsetofpointswhereHf\Xisnotsmoothofdimensionm�1.AlthoughTheorem1.1showsthatforanonemptysmoothquasiprojectivesubschemeXPnofdimensionm0,thereisapositiveprobabilitythat(Hf\X)sing6=;,wenowshowthattheprobabilitythatdim(Hf\X)sing1iszero.Theorem3.2.LetXbeasmoothquasiprojectivesubschemeofPnofdimensionm0overFq.DeneS:=ff2Shomog:dim(Hf\X)sing1g:Then(S)=0.Proof.ThisisacorollaryofLemma2.6withU=X,sinceSQhigh.Remark.Iff2ShomogissuchthatHfisnotgeometricallyintegralofdimensionn�1,thendim(Hf)singn�2.HenceTheorem3.2withX=PngivesanewproofofProposition2.7,atleastwhenn3.3.3.Space-llingcurves.WenextuseTheorem1.2toanswerarmativelyalltheopenquestionsin[Kat99].Intheirstrongestforms,theseareQuestion10:GivenasmoothprojectivegeometricallyconnectedvarietyXofdimensionm2overFq,andaniteextensionEofFq,istherealwaysaclosedsubschemeYinX,Y6=X,suchthatY(E)=X(E)andsuchthatYissmoothandgeometricallyconnectedoverFq? 10BJORNPOONENQuestion13:GivenaclosedsubschemeXPnoverFqthatissmoothandgeometricallyconnectedofdimensionm,andapointP2X(Fq),isittrueforalld1thatthereexistsahypersurfaceHPnofdegreedsuchthatPliesonHandH\Xissmoothofdimensionm�1?Bothofthesequestionsareansweredbythefollowing:Theorem3.3.LetXbeasmoothquasiprojectivesubschemeofPnofdimensionm1overFq,andletFXbeanitesetofclosedpoints.ThenthereexistsasmoothprojectivegeometricallyintegralhypersurfaceHPnsuchthatH\Xissmoothofdimensionm�1andcontainsF.Remarks.(1)Ifm2andifXinTheorem3.3isgeometricallyconnectedandprojectiveinadditiontobeingsmooth,thenH\Xwillbegeometricallyconnectedandprojectivetoo.ThisfollowsfromCorollaryIII.7.9in[Har77].(2)Recallthatifavarietyisgeometricallyconnectedandsmooth,thenitisgeometricallyintegral.(3)Question10and(partially)Question13wereindependentlyansweredbyGabber[Gab01].ProofofTheorem3.3.LetTP;XbetheZariskitangentspaceofapointPonX.AteachP2Fchooseacodimension1subspaceVPTP;PnnotequaltoTP;X.WewillapplyTheorem1.3withthefollowinglocalconditions:forP2F,UPistheconditionthatthehypersurfaceHfpassesthroughPandTP;H=VP;forP62F,UPistheconditionthatHfandHf\Xbesmoothofdimensionsn�1andm�1,respectively,atP.Theorem1.3(withProposition2.7)impliestheexistenceofasmoothprojectivegeometricallyintegralhypersurfaceHPnsatisfyingtheseconditions.Remark.IfwedidnotinsistinTheorem3.3thatHbesmooth,thenintheproof,Theorem1.2wouldsuceinplaceofTheorem1.3.ThisweakenedversionofTheorem3.3isalreadyenoughtoimplyCorollaries3.4and3.5,andTheorem3.7.Corollary3.6alsofollowsfromTheorem1.2.Corollary3.4.LetXbeasmooth,projective,geometricallyintegralvarietyofdimensionm1overFq,letFbeanitesetofclosedpointsofX,andletybeanintegerwith1ym.Thenthereexistsasmooth,projective,geometricallyintegralsubvarietyYXofdimensionysuchthatFY.Proof.UseTheorem3.3withreverseinductionony.Corollary3.5(Space-llingcurves).LetXbeasmooth,projective,geometricallyintegralvarietyofdimensionm1overFq,andletEbeaniteextensionofFq.Thenthereexistsasmooth,projective,geometricallyintegralcurveYXsuchthatY(E)=X(E).Proof.ApplyCorollary3.4withy=1andFthesetofclosedpointscorrespondingtoX(E).Inasimilarway,weprovethefollowing:Corollary3.6(Space-avoidingvarieties).LetXbeasmooth,projective,geometricallyin-tegralvarietyofdimensionmoverFq,andlet`andybeintegerswith`1and1ym.Thenthereexistsasmooth,projective,geometricallyintegralsubvarietyYXofdimensionysuchthatYhasnopointsofdegreelessthan`. BERTINITHEOREMSOVERFINITEFIELDS11Proof.RepeattheargumentsusedintheproofofTheorem3.3andCorollary3.4,butintherstapplicationofTheorem1.3,insteadforcethehypersurfacetoavoidthenitelymanypointsofXofdegreelessthan`.3.4.Albanesevarieties.Forasmooth,projective,geometricallyintegralvarietyXoveraeld,letAlbXbeitsAlbanesevariety.Aspointedoutin[Kat99],apositiveanswertoQuestion13impliesthateverypositivedimensionalabelianvarietyAoverFqcontainsasmooth,projective,geometricallyintegralcurveYsuchthatthenaturalmapAlbY!Aissurjective.Wegeneralizethisslightlyinthenextresult,whichstrengthensTheorem11of[Kat99]intheniteeldcase.Theorem3.7.LetXbeasmooth,projective,geometricallyintegralvarietyofdimensionm1overFq.Thenthereexistsasmooth,projective,geometricallyintegralcurveYXsuchthatthenaturalmapAlbY!AlbXissurjective.Proof.Chooseaprime`notequaltothecharacteristic.Representeach`-torsionpointin(AlbX)( Fq)byazero-cycleofdegreezeroonX,andletFbethenitesetofclosedpointsappearinginthese.UseCorollary3.4toconstructasmooth,projective,geometricallyintegralcurveYpassingthroughallpointsofF.TheimageofAlbY!AlbXisanabeliansubvarietyofAlbXcontainingallthe`-torsionpoints,sotheimageequalsAlbX.(Thetrickofusingthe`-torsionpointsisduetoGabber[Kat99].)Remarks.(1)AslightlymoregeneralargumentprovesTheorem3.7overanarbitraryeldk[Gab01,Proposition2.4].(2)ItisalsotruethatanyabelianvarietyoveraeldkcanbeembeddedasanabeliansubvarietyoftheJacobianofasmooth,projective,geometricallyintegralcurveoverk[Gab01].3.5.Planecurves.TheprobabilitythataprojectiveplanecurveoverFqisnonsingularequalsP2(3)�1=(1�q�1)(1�q�2)(1�q�3):(WeinterpretthisprobabilityasthedensitygivenbyTheorem1.1forX=P2inP2.)Theorem1.3withasimplelocalcalculationshowsthattheprobabilitythataprojectiveplanecurveoverFqhasatworstnodesassingularitiesequalsP2(4)�1=(1�q�2)(1�q�3)(1�q�4):ForF2,theseprobabilitiesare21=64and315=512.Remark.AlthoughTheorem1.1guaranteestheexistenceofasmoothplanecurveofdegreedoverFqonlywhendissucientlylargerelativetoq,infactsuchacurveexistsforeveryd1andeveryniteeldFq.Moreover,thecorrespondingstatementforhypersurfacesofspecieddimensionanddegreeistrue[KS99,x11.4.6].Infact,foranyeldkandintegersn1,d3with(n;d)notequalto(1;3)or(2;4),thereexistsasmoothhypersurfaceXoverkofdegreedinPn+1suchthatXhasnonontrivialautomorphismsover k[Poo00].Thislaststatementisfalsefor(1;3);whetherornotitholdsfor(2;4)isanopenquestion. 12BJORNPOONEN4.AnopenquestionInresponsetoTheorem1.1,MattBakerhasaskedthefollowing:Question4.1.FixasmoothquasiprojectivesubschemeXofdimensionmoverFq.Doesthereexistanintegern0�0suchthatfornn0,if:X!PnisanembeddingsuchthatnoconnectedcomponentofXismappedbyintoahyperplaneinPn,thenthereexistsahyperplaneHPnoverFqsuchthatH\(X)issmoothofdimensionm�1?Theorem1.1provesthattheanswerisyes,ifoneallowsonlytheembeddingsobtainedbycomposingaxedinitialembeddingX!Pnwithd-upleembeddingsPn!PN.Never-theless,weconjecturethatforeachXofpositivedimension,theanswertoQuestion4.1isno.5.AnarithmeticanalogueWeformulateananalogueofTheorem1.1inwhichthesmoothquasiprojectiveschemeXoverFqisreplacedbyaregularquasiprojectiveschemeXoverSpecZ,andweseekhyperplanesectionsthatareregular.ThereasonforusingregularityinsteadofthestrongerconditionofbeingsmoothoverZisdiscussedinSection5.7.Fixn2N=Z0.RedeneSasthehomogeneouscoordinateringZ[x0;:::;xn]ofPnZ,letSdSbetheZ-submoduleofhomogeneouspolynomialsofdegreed,andletShomog=S1d=0Sd.Ifpisprime,letSd;pbethesetofhomogeneouspolynomialsinFp[x0;:::;xn]ofdegreed.Foreachf2Sd,letHfbethesubschemeProj(S=(f))PnZ.Similarly,forf2Sd;p,letHfbeProj(Fp[x0;:::;xn]=(f))PnFp.IfPisasubsetofZNforsomeN1,denetheupperdensity (P):=maxlimsupB(1)!1


limsupB(N)!1#(P\Box) #Box;whererangesoverpermutationsoff1;2;:::;NgandBox=f(x1;:::;xN)2ZN:jxijBiforallig:(Inotherwords,wetakethelimsuponlyovergrowingboxeswhosedimensionscanbeorderedsothateachisverylargerelativetothepreviousdimensions.)Denelowerdensity (P)similarlyusingminandliminf.Deneupperandlowerdensities dand dofsubsetsofaxedSdbyidentifyingSdwithZNusingaZ-basisofmonomials.IfPShomog,dene (P)=limsupd!1 d(P\Sd)and (P)=liminfd!1 d(P\Sd).Finally,ifPisasubsetofShomog,dene(P)asthecommonvalueof (P)and (P)if (P)= (P).Thereasonforchoosingthisdenitionisthatitmakesourproofwork;aesthetically,wewouldhavepreferredtoproveastrongerstatementbydeningdensityasthelimitoverarbitraryboxesinSdwithminfd;B1;:::;BNg!1;probablysuchastatementisalsotruebutextremelydiculttoprove.ForaschemeXofnitetypeoverZ,denethezetafunction[Ser65,x1.3]X(s):=YclosedP2X�1�#(P)�s
�1;where(P)isthe(nite)residueeldofP.ThisgeneralizesthedenitionofSection1,sinceaschemeofnitetypeoverFqcanbeviewedasaschemeofnitetypeoverZ.Ontheotherhand,SpecZ(s)istheRiemannzetafunction. BERTINITHEOREMSOVERFINITEFIELDS13Theabcconjecture,formulatedbyD.MasserandJ.OesterleinresponsetoinsightsofR.C.Mason,L.Szpiro,andG.Frey,isthestatementthatforany�0,thereexistsaconstantC=C()�0suchthatifa;b;carecoprimepositiveintegerssatisfyinga+b=c,thencC(Ypjabcp)1+.Forconvenience,wesaythataschemeXofnitetypeoverZisregularofdimensionmifforeveryclosedpointPofX,thelocalringOX;Pisregularofdimensionm.ForaschemeXofnitetypeoverZ,thisisequivalenttotheconditionthatOX;PberegularforallP2XandallirreduciblecomponentsofXhaveKrulldimensionm.IfXissmoothofrelativedimensionm�1overSpecZ,thenXisregularofdimensionm,buttheconverseneednothold.Theemptyschemeisregularofeverydimension.Theorem5.1(Bertiniforarithmeticschemes).AssumetheabcconjectureandConjec-ture5.2below.LetXbeaquasiprojectivesubschemeofPnZthatisregularofdimensionm0.DeneP:=ff2Shomog:Hf\Xisregularofdimensionm�1g:Then(P)=X(m+1)�1.Remark.ThecaseX=P0Z=SpecZinP0ZofTheorem5.1isthestatementthatthedensityofsquarefreeintegersis(2)�1,whereistheRiemannzetafunction.TheproofofTheorem5.1ingeneralwillinvolvequestionsaboutsquarefreevaluesofmultivariablepolynomials.GivenaschemeX,letXQ=XQ,andletXp=XFpforeachprimep.Conjecture5.2.LetXbeanintegralquasiprojectivesubschemeofPnZthatissmoothoverZofrelativedimensionr.Thereexistsc&#x-277;0suchthatifdandparesucientlylarge,then#ff2Sd;p:dim(Hf\Xp)sing1g #Sd;pc p2:HeuristicallyoneexpectsthatConjecture5.2istrueevenifc=p2isreplacedbyc=pkforanyxedk2.Ontheotherhand,fortheapplicationtoTheorem5.1,itwouldsucetoproveaweakformofConjecture5.2withtheupperboundc=p2replacedbyanyp�0suchthatPpp1.Weusedc=p2onlytosimplifythestatement.Ifdissucientlylargerelativetop,thenTheorem3.2providesasuitableupperboundontheratioinConjecture5.2.Ifpissucientlylargerelativetod,thenonecanderiveasuitableupperboundfromtheWeilConjectures.(Inparticular,thetruthofConjecture5.2isunchangedifwedroptheassumptionthatdandparesucientlylarge.)Thedicultyliesinthecasewherediscomparabletop.SeeSection5.4,foraproofofConjecture5.2inthecasewheretheclosureofXQinPnQhasatmostisolatedsingularities.5.1.Singularpointswithsmallresidueeld.WebegintheproofofTheorem5.1withanaloguesofresultsinSection2.1.IfMisaniteabeliangroup,letlengthZMbeitslengthasaZ-module.Lemma5.3.IfYisazero-dimensionalclosedsubschemeofPnZ,thenthemapd:Sd=H0(PnZ;O(d))!H0(Y;OY(d))issurjectivefordlengthZH0(Y;OY)�1. 14BJORNPOONENProof.AssumedlengthZH0(Y;OY)�1.ThecokernelCofdisnite,sinceitisaquotientofthenitegroupH0(Y;OY(d)).Moreover,Chastrivialp-torsionforeachprimep,byLemma2.1appliedtoYFpinPnFp.ThusC=0.Hencedissurjective.Lemma5.4.IfPZNisaunionofcdistinctcosetsofasubgroupGZNofindexm,then(P)=c=m.Proof.Withoutlossofgenerality,wemayreplaceGwithitssubgroup(mZ)Nofniteindex.Theresultfollows,sinceanyoftheboxesinthedenitionofcanbeapproximatedbyaboxofdimensionsthataremultiplesofm,withanerrorthatbecomesnegligiblecomparedwiththenumberoflatticepointsintheboxastheboxdimensionstendtoinnity.IfXisaschemeofnitetypeoverZ,deneXasthesetofclosedpointsPwith#(P)r.(Thiscon ictswiththecorrespondingdenitionbeforeLemma2.2;forgetthatone.)DeneXrsimilarly.WesaythatXisregularofdimensionmataclosedpointPofPnZifeitherP62XorOX;Pisaregularlocalringofdimensionm.Lemma5.5(Smallsingularities).LetXbeaquasiprojectivesubschemeofPnZthatisregularofdimensionm0.DenePr:=ff2Shomog:Hf\Xisregularofdimensionm�1atallP2Xg:Then(Pr)=YP2X�1�#(P)�(m+1)
:Proof.GivenLemmas5.3and5.4,theproofisthesameasthatofLemma2.2withZ=;.5.2.Reductions.Theorem1of[Ser65]showsthatYP2X�1�#(P)�(m+1)
convergestoX(m+1)�1asr!1.ThusTheorem5.1followsfromLemma5.5andthefollowing,whoseproofwilloccupytherestofSection5.Lemma5.6(Largesingularities).AssumetheabcconjectureandConjecture5.2.LetXbeaquasiprojectivesubschemeofPnZthatisregularofdimensionm0.DeneQlarger:=ff2Shomog:thereexistsP2XrsuchthatHf\Xisnotregularofdimensionm�1atPg:Thenlimr!1 (Qlarger)=0.Lemma5.6holdsforXifitholdsforeachsubschemeinanopencoverofX,sincebyquasicompactnessanysuchopencoverhasanitesubcover.Inparticular,wemayassumethatXisconnected.SinceXisalsoregular,Xisintegral.IftheimageofX!SpecZisaclosedpoint(p),thenXissmoothofdimensionmoverFp,andLemma5.6forXfollowsfromLemmas2.4and2.6.Thusfromnowon,weassumethatXdominatesSpecZ.SinceXisregular,itsgenericberXQisregular.SinceQisaperfecteld,itfollowsthatXQissmoothoverQ,ofdimensionm�1.By[EGAIV(4),17.7.11(iii)],thereexistsanintegert1suchthatXZ[1=t]issmoothofrelativedimensionm�1overZ[1=t]. BERTINITHEOREMSOVERFINITEFIELDS155.3.Singularpointsofsmallresiduecharacteristic.Lemma5.7(Singularitiesofsmallcharacteristic).Fixanonzeroprimep2Z.LetXbeanintegralquasiprojectivesubschemeofPnZthatdominatesSpecZandisregularofdimensionm0.DeneQp;r:=ff2Shomog:thereexistsP2Xpwith#(P)rsuchthatHf\Xisnotregularofdimensionm�1atPg:Thenlimr!1 (Qp;r)=0.Proof.WemayassumethatXpisnonempty.Then,sinceXpiscutoutinXbyasingleequationp=0,andsincepisneitheraunitnorazerodivisorinH0(X;OX),dimXp=m�1.LetQmediump;r:=[d0ff2Sd:thereexistsP2Xpwithr#(P)pd=(m+1)suchthatHf\Xisnotregularofdimensionm�1atPgandQhighp:=[d0ff2Sd:thereexistsP2Xpwith#(P)�pd=(m+1)suchthatHf\Xisnotregularofdimensionm�1atPg:SinceQp;r=Qmediump;r[Qhighp,itsucestoprovelimr!1 (Qmediump;r)=0and (Qhighp)=0.WewilladapttheproofsofLemmas2.4and2.6.IfPisaclosedpointofX,letmX;POXbetheidealsheafcorrespondingtoP,andletYPbetheclosedsubschemeofXcorrespondingtotheidealsheafm2X;P.Forxedd,thesetQmediump;r\SdiscontainedintheunionoverPwithr#(P)pd=(m+1)ofthekerneloftherestrictionP:Sd!H0(YP;O(d)).SinceH0(YP;O(d))'H0(YP;OYP)haslength(m+1)[(P):Fp]dasaZ-module,PissurjectivebyLemma5.3,andLemma5.4implies (kerP)=#(P)�(m+1).Thus (Qmediump;r\Sd)XP (kerP)=XP#(P)�(m+1):wherethesumisoverP2Xpwithr#(P)pd=(m+1).Thecrudeform#Xp(Fpe)=O(pe(m�1))oftheboundin[LW54]impliesthatlimr!1 (Qmediump;r)=limr!1limd!1 (Qmediump;r\Sd)=0:NextweturntoQhighp.SincewearefreetopasstoanopencoverofX,wemayassumethatXiscontainedinthesubsetAnZ:=fx06=0gofPnZ.LetA=Z[x1;:::;xn]betheringofregularfunctionsonAnZ.IdentifySdwiththesetofdehomogenizationsAd=ff2A:degfdg,wheredegfdenotestotaldegree.Let bethesheafofdierentials Xp=Fp.ForP2Xp,denethedimensionoftheber(P)=dim(P) OXp(P): 16BJORNPOONENLetmXp;PbethemaximalidealofthelocalringOXp;P.IfPisaclosedpointofXp,theisomorphism OXp(P)'mXp;P m2Xp;PofPropositionII.8.7of[Har77]showsthat(P)=dim(P)mXp;P=m2Xp;P;moreoverpOX;P!mX;P m2X;P!mXp;P m2Xp;P!0isexact.SinceXisregularofdimensionm,themiddletermisa(P)-vectorspaceofdimensionm.Butthemoduleontheleftisgeneratedbyoneelement.Hence(P)equalsm�1ormateachclosedpointP.LetY=fP2Xp:(P)mg.ByExerciseII.5.8(a)of[Har77],Yisaclosedsubset,andwegiveYthestructureofareducedsubschemeofXp.LetU=Xp�Y.ThusforclosedpointsP2Xp,(P)=(m�1;ifP2Um;ifP2Y.IfUisnonempty,thendimU=dimXp=m�1,soUissmoothofdimensionm�1overFp,and jUislocallyfree.AtaclosedpointP2U,wecanndt1;:::;tn2Asuchthatdt1;:::;dtm�1representanOXp;P-basisforthestalk P,anddtm;:::;dtnrepresentabasisforthekernelof An=Fp OXp;P! P.Let@1;:::;@n2TAn=Fp;Pbethebasisofderivationsdualtodt1;:::;dtn.Chooses2AnonvanishingatPsuchthats@iextendstoaglobalderivationDi:A!Afori=1;2;:::;m�1.InsomeneighborhoodVofPinAnFp,dt1;:::;dtnformabasisof V=Fp,anddt1;:::;dtm�1formabasisof U\V=Fp,ands2O(V).Bycompactness,wemaypasstoanopencoverofXtoassumeUV.IfHf\XisnotregularataclosedpointQ2U,thentheimageoffinmU;Q=m2U;Qmustbezero,anditfollowsthatD1f,...,Dm�1f,fallvanishatQ.Thesetoff2SdsuchthatthereexistssuchapointinUcanbeboundedusingtheinductionargumentintheproofofLemma2.6.Itremainstoboundthef2SdsuchthatHf\XisnotregularatsomeclosedpointP2Y.SinceYisreduced,andsincethebersofthecoherentsheaf OYonYallhavedimensionm,thesheafislocallyfreebyExerciseII.5.8(c)in[Har77].Bythesameargumentasintheprecedingparagraph,wecanpasstoanopencoverofX,andndt1;:::;tn;s2Asuchthatdt1;:::;dtnareabasisoftherestrictionof An=FptoaneighborhoodofYinAnFp,anddt1;:::;dtmareanOY-basisof OY,ands2O(Y)issuchthatif@1;:::;@nisthedualbasistodt1;:::;dtn,thens@iextendstoaderivationDi:A!Afori=1;:::;m�1.(WecouldalsodeneDifori=m,butwealreadyhaveenough.)WenishagainbyusingtheinductionargumentintheproofofLemma2.6.5.4.Singularpointsofmidsizedresiduecharacteristic.Whileexaminingpointsoflargerresiduecharacteristic,wemaydeletethebersabovesmallprimesofZ.Henceinthissectionandthenext,ourlemmaswillsupposethatXissmoothoverZ.Lemma5.8(Singularitiesofmidsizedcharacteristic).AssumeConjecture5.2.LetXbeanintegralquasiprojectivesubschemeofAnZthatdominatesSpecZandissmoothoverZof BERTINITHEOREMSOVERFINITEFIELDS17relativedimensionm�1.Ford;L;M1,deneQd;L
:=ff2Sd:thereexistpsatisfyingLpMandP2XpsuchthatHf\Xisnotregularofdimensionm�1atPg:Given&#x-278;0,ifdandLaresucientlylarge,then (Qd;L
).Proof.IfPisaclosedpointofdegreeatmostd=(m+1)overFpwhereLpM,thenthesetoff2SdsuchthatHf\Xisnotregularofdimensionm�1atPhasupperdensity#(P)�(m+1),asintheargumentforQmediump;rinLemma5.7.Thesumover#(P)�(m+1)overallsuchPissmallifLissucientlylarge:thisfollowsfrom[LW54],asusual.ByConjecture5.2,theupperdensityofthesetoff2SdsuchthatthereexistspwithLpMsuchthatdim(Hf\Xp)sing1isboundedbyPLc=p2,whichagainissmallifLissucientlylarge.LetEd;pbethesetoff2Sdforwhich(Hf\Xp)singisniteandHf\Xfailstoberegularofdimensionm�1atsomeclosedpointP2Xpofdegreegreaterthand=(m+1)overFp.ItremainstoshowthatifdandLaresucientlylarge,PL (Ed;p)issmall.Writef=f0+pf1wheref0hascoecientsinf0;1;:::;p�1g.Oncef0isxed,(Hf\Xp)singisdetermined,andinthecasewhereitisnite,weletP1;:::;P`beitsclosedpointsofdegreegreaterthand=(m+1)overFp.NowHf\Xisnotregularofdimensionm�1atPiifandonlyiftheimageoffinOX;Pi=m2X;Piiszero;forxedf0,thisisaconditiononlyontheimageoff1inOXp;Pi=mXp;Pi.ItfollowsfromLemma2.5thatthefractionoff1forthisholdsisatmostp�where=d=(m+1).Thus (Ed;p)`p�.Asusual,wemayassumewehavereducedtothecasewhere(Hf\Xp)singiscutoutbyD1f;:::;Dm�1f;fforsomederivationsDi,andhencebyBezout'sTheorem,`=O(dm)=O(p�2)asd!1,so (Ed;p)=O(p�2).HencePL (Ed;p)issmallwheneverdandLarelarge.Thefollowinglemmaanditsproofweresuggestedbythereferee.Lemma5.9.Conjecture5.2holdswhentheclosure XQofXQinPnQhasatmostisolatedsingularities.Proof.Weuseinductiononn.Let XbetheclosureofXinPnZ.Since XQhasatmostisolatedsingularities,alinearchangeofcoordinatesoverQmakes XQ\fx0=0gissmoothofdimensionr�1.SincethestatementofConjecture5.2isunchangedbydeletingbersofX!SpecZabovesmallprimes,wemayassumethat X\fx0=0gissmoothoverZofrelativedimensionr�1.Next,wemayenlargeXtoassumethatXisthesmoothlocusof X!SpecZ,sincethisonlymakesthedesiredconclusionhardertoprove.ThesmoothZ-scheme X\fx0=0giscontainedinthesmoothlocusXoftheZ-scheme X,soX\fx0=0g= X\fx0=0g.Iff2Sd;pissuchthatdim(Hf\Xp)sing1,thentheclosureof(Hf\Xp)singintersectsfx0=0g.ButX\fx0=0g= X\fx0=0g,so(Hf\Xp)singitselfintersectsfx0=0g.Thusitsucestoprove#ff2Sd;p:(Hf\Xp)sing\fx0=0g6=;g #Sd;pc p2:Foraclosedpointyofdegreed=(r+1)ofXp\fx0=0g,theprobabilitythaty2(Hf\Xp)singis#(y)�r�1andthesumoversuchpointsistreatedasintheproofofLemma5.8. 18BJORNPOONENItremainstocountf2Sd;psuchthatHf\Xpissingularataclosedpointyofdegree�d=(r+1)ofXp\fx0=0g.Notethat(Hf\Xp)sing\fx0=0giscontainedinthesubschemef:=(Hf\Xp\fx0=0g)sing.BytheinductivehypothesisappliedtoX\fx0=0g,wemayrestrictthecounttothefforwhichHf\Xp\fx0=0gisofpuredimensionr�2andfisnite.ThenbyBezout,#f=O(dr),wheretheimpliedconstantdependsonlyonX.Ifwewritef=f0+f1x0+f2x20+:::withfi2Fp[x1;:::;xn],thenfdependsonlyonf0.Forxedf0andy2f=f0,whetherornoty2(Hf\Xp)singdependsonlyonthe\value"off1aty(whichisin�(y;O(d�1)jy)),andatmostonevaluecorrespondstoasingularity.TheFp-vectorspaceofpossiblevaluesoff1atyhasdimensionmin(deg(y);d),soifwerestricttoyofdegree�d=(r+1),theprobabilitythaty2(Hf\Xp)singisatmostp�d=(r+1).Thus,forxedf0,theprobabilitythatHf\XpissingularatsomesuchyisO(drp�d=(r+1)),whichisO(p�2)fordlargeenough.Finally,theimpliedconstantisindependentoff0,sotheoverallprobabilityisagainO(p�2).5.5.Singularpointsoflargeresiduecharacteristic.Wecontinuetoidentifyhomoge-neouspolynomialsinx0;:::;xnwiththeirdehomogenizationsobtainedbysettingx0,whenneededtoconsiderthemasfunctionsonAnZPnZ.Lemma5.10.LetXbeanintegralquasiprojectivesubschemeofAnZthatdominatesSpecZandissmoothoverZofrelativedimensionm�1.Fixd1.Letf2Z[c0;:::;cN][x0;:::;xn]bethegenerichomogeneouspolynomialinx0;:::;xnoftotaldegreed,havingtheindeter-minatesc0;:::;cNascoecients(soN+1isthenumberofhomogeneousmonomialsinx0;:::;xnoftotaldegreed).ThenthereexistsanintegerM�0andasquarefreepoly-nomialR(c0;:::;cN)2Z[c0;:::;cN]suchthatiffisobtainedfromfbyspecializingthecoecientscitointegers i,andifHf\XfailstoberegularataclosedpointintheberXpforsomeprimepM,thenp2dividesthevalueR( 0;:::; N).Proof.Byusinga\d-upleembedding"ofX(i.e.,mappingAntoANusingallhomogeneousmonomialsinx0;:::;xnoftotaldegreed),wereducetothecaseofintersectingXinsteadwithananehyperplaneHfAnZdenedby(thedehomogenizationof)f=c0x0+


+cnxn.LetAn+1=An+1Zbetheanespacewhosepointscorrespondtosuchhomogeneouslinearforms.Thusc0;:::;cnarethecoordinatesonAn+1.IfXhasrelativedimensionnoverSpecZ(soXisanonemptyopensubsetofAn),wemaytriviallytakeR=c0ifn=0andR=c0c1ifn�0.Thereforeweassumethattherelativedimensionisstrictlylessthanninwhatfollows.LetXAn+1bethereducedclosedsubschemeofpoints(x;f)suchthatthevarietyHf\Xovertheresidueeldof(x;f)isnotsmoothofdimensionm�2atx.Then,becausewehaveexcludedthedegeneratecaseofthepreviousparagraph,QistheclosureinXQAn+1QoftheinverseimageunderXQAn+1Q99KXQPnQoftheconormalvarietyCXXQPnQasdenedin[Kle86,I-2](underslightlydierenthypotheses).Concretely,isthesubschemeofXAn+1locallycutoutbytheequationsD1f=


Dm�1f=f=0wheretheDiaredenedlocallyonXasinthepenultimateparagraphoftheproofofLemma5.7.LetIbethescheme-theoreticimageofundertheprojection:!An+1.ThusIQAn+1QistheconeoverthedualvarietyX,denedasthescheme-theoreticimageofthecorrespondingprojectionCX!PnQ.By[Kle86,p.168],wehavedimCX=n�1,sodimXn�1. BERTINITHEOREMSOVERFINITEFIELDS19Case1.dimX=n�1.ThenIQisanintegralhypersurfaceinAnQ,saygivenbytheequationR0(c0;:::;cn)=0,whereR0isanirreduciblepolynomialwithcontent1.AfterinvertinganitenumberofnonzeroprimesofZ,wemayassumethatR0=0isalsotheequationdeningIinAnZ.ChooseMgreaterthanalltheinvertedprimes.SincedimX=n�1,theprojectionCX!Xisabirationalmorphism.Byduality(seetheMonge-Segre-Wallacecriteriononp.169of[Kle86]),CX=CX,soCX!Xisabirationalmap.Itfollowsthat:!Iisabirationalmorphism.ThuswemaychooseanopendensesubsetI0ofIsuchthatthebirationalmorphism:!Iinducesanisomorphism0!I0,where0=�1(I0).ByHilbert'sNullstellensatz,thereexistsR12Z[c0;:::;cn]suchthatR1vanishesontheclosedsubsetI�I0butnotonI.WemayassumethatR1issquarefree.DeneR=R0R1.ThenRissquarefree.SupposethatHf\XfailstoberegularatapointP2XpwithpM.Let betheclosedpointofAn+1denedbyc0� 0=


=cn� n=p=0.Thenthepoint(P; )ofXAn+1isin.Hence 2I,soR0( 0;:::; n)isdivisiblebyp.If 2I�I0,thenR1( 0;:::; n)isdivisiblebypaswell,soR( 0;:::; n)isdivisiblebyp2,asdesired.Thereforeweassumefromnowonthat 2I0,so(P; )20.LetWbetheinverseimageofI0undertheclosedimmersionSpecZ!An+1denedbytheideal(c0� 0;:::;cn� n).LetVbetheinverseimageof0underthemorphismX,!XAn+1inducedbythepreviousclosedimmersion.Thuswehaveacubeinwhichthetop,bottom,front,andbackfacesarecartesian:V//  ##GGGGGGGGGG0 %%JJJJJJJJJJX//  XAn+1 W// ##GGGGGGGGGI0$$JJJJJJJJJJSpecZ// An+1Near(P; )2XAn+1thefunctionsD1f;


;Dm�1f;fcutout(andhencealsoitsopensubset0)locallyinXAn+1.ThenOV;P=OX;P=(D1f;D2f;:::;Dm�1f;f).Byassumption,Hf\XisnotregularatP,sofmapstozeroinOX;P=m2X;P.NowOX;Pisaregularlocalringofdimensionm,Dif2mX;P,andf2m2X;P,sothequotientOV;Phaslengthatleast2.Since0!I0isanisomorphism,thecubeshowsthatV!Wisanisomorphismtoo.HencethelocalizationofWatphaslengthatleast2.OntheotherhandI0isanopensubschemeofI,whoseidealisgeneratedbyR0(c0;:::;cn)(aftersomeprimeswereinverted),soWisanopensubschemeofZ=(R0( 1;:::; N)).ThusR0( 0;:::; n)isdivisiblebyp2atleast.ThusR( 0;:::; n)isdivisiblebyp2.Case2.dimXn�1.ThenIQisofcodimension2inAn+1Q.Invertingnitelymanyprimesifnecessary,wecanndapairofdistinctirreduciblepolynomialsR1;R22Z[c0;:::;cn]vanishingonI.LetR=R1R2.AsinCase1,ifHf\XfailstoberegularatP2XpwithpM,thenthevaluesofR1andR2bothvanishmodulop,sothevalueofRisdivisiblebyp2. 20BJORNPOONENBecauseofLemma5.10,wewouldliketoknowthatmostvaluesofamultivariablepoly-nomialoverZarealmostsquarefree(thatis,squarefreeexceptforprimefactorslessthanM).Itisherethatweneedtoassumetheabcconjecture.Theorem5.11(Almostsquarefreevaluesofpolynomials).Assumetheabcconjecture.LetF2Z[x1;:::;xn]besquarefree.ForM�0,deneSM:=f(a1;:::;an)2ZnjF(a1;:::;an)isdivisiblebyp2forsomeprimepMg:Then (SM)!0asM!1.Proof.Then=1caseisin[Gra98].ThegeneralcasefollowsfromLemma6.2of[Poo03],inthesamewaythatCorollary3.3therefollowsfromTheorem3.2there.Lemma6.2thereisprovedtherebyreductiontothen=1case.Remarks.(1)Theseresultsassumetheabcconjecture,butthespecialcasewhereFfactorsintoone-variablepolynomialsofdegree3isknownunconditionally[Hoo67].Otherunconditionalresultsarecontainedin[GM91].(2)Theorem5.11togetherwithasimplesieveletsoneshowthatthenaiveheuris-tic(multiplyingprobabilitiesforeachprimep)correctlypredictsthedensityof(a1;:::;an)2ZnforwhichF(a1;:::;an)issquarefree,assumingtheabcconjecture.Lemma5.12(Singularitiesoflargecharacteristic).Assumetheabcconjecture.LetXbeanintegralquasiprojectivesubschemeofAnZthatdominatesSpecZandissmoothoverZofrelativedimensionm�1.DeneQd;M:=ff2Sd:thereexistspMandP2XpsuchthatHf\Xisnotregularofdimensionm�1atPg:Ifdissucientlylarge,thenlimM!1 (Qd;M)=0.Proof.WemayassumethatdislargeenoughforLemma5.10.ApplyTheorem5.11tothesquarefreepolynomialRprovidedbyLemma5.10forX.5.6.Endofproof.WearenowreadytoproveTheorem5.1.RecallthatinSection5.2wereducedtotheproblemofprovingLemma5.6inthecasewhereXisanintegralquasipro-jectivesubschemeofAnZsuchthatXdominatesSpecZandisregularofdimensionm0.InLemma5.6,dtendstoinnityforeachxedr,andthenrtendstoinnity.WechooseLdependingonr,andMdependingonrandd,suchthat1LrdM.(Thepreciserequirementimpliedbyeachiswhateverisneededbelowfortheapplicationsofthelemmasbelow.)Then(1)Qlarger\Sd [pL(Qp;r\Sd)![Qd;L
[Qd;M;andwewillboundtheupperdensityofeachtermontheright.RecallfromtheendofSection5.2thatXhasasubschemeoftheformX0=XSpecZ[1=t]thatissmoothoverZ.WemayassumeL&#xM]TJ;&#x/F14;&#x 11.;镒&#x Tf ;.7;# 1;&#x.793;&#x Td ;&#x[000;t.ByLemma5.7,limr!1 (Qp;r)=0foreachp,so SpLQp;rissmall(bywhichwemeantendingtozero)ifrsucientlylargerelativetoL.ByLemma5.8appliedtoX0,ifLanddaresucientlylarge,then (Qd;L
)issmall.ByLemma5.12appliedtoX0,ifdissucientlylarge,andMissucientlylargerelativetod,then (Qd;M) BERTINITHEOREMSOVERFINITEFIELDS21issmall.Thusby(1), (Qlarger)issmallwheneverrislargeanddissucientlylargerelativetor.ThiscompletestheproofofLemma5.6andhenceofTheorem5.1.Remark.ArithmeticanaloguesofTheorems1.2and1.3,andofmanyoftheapplicationsinSection3canbeprovedaswell.5.7.Regularversussmooth.OnemightaskwhathappensinTheorem5.1ifweaskforHf\Xtobenotonlyregular,butalsosmoothoverZ.Wenowshowunconditionallythatthisrequirementissostrict,thatatmostadensityzerosubsetofpolynomialsfsatisesit,eveniftheoriginalschemeXissmoothoverZ.Theorem5.13.LetXbeanonemptyquasiprojectivesubschemeofPnZthatissmoothofrelativedimensionm0overZ.DenePsmooth:=ff2Shomog:Hf\Xissmoothofrelativedimensionm�1overZg:Then(Psmooth)=0.Proof.LetPsmoothr:=ff2Shomog:Hf\Xissmoothofrelativedimensionm�1overZatallP2Xg:SupposeP2Xliesabovetheprime(p)2SpecZ.LetYbetheclosedsubschemeofXpcorrespondingtotheidealsheafm2wheremistheidealsheafoffunctionsonXpvanishingatP.Thenforf2Sd,Hf\Xissmoothofrelativedimensionm�1overZatPifandonlyiftheimageoffinH0(Y;O(d))isnonzero.ApplyingLemma5.3totheunionofsuchYoverallP2X,andusing#H0(Y;O(d))=#(P)m+1,wend(Psmoothr)=YP2X�1�#(P)�(m+1)
:SincedimX=m+1,X(s)hasapoleats=m+1andourproductdivergesto0asr!1.(SeeTheorems1and3(a)in[Ser65].)ButPsmoothPsmoothrforallr,so(Psmooth)=0.AdensityzerosubsetofShomogcanstillbenonemptyoreveninnite.Forexample,ifX=SpecZ[1=2;x],!P1Z,thenPsmooth\Sdisinniteforinnitelymanyd:Hf\XissmoothoverZwheneverfisthehomogenizationof(x�a)2b�2forsomea;b2Zwithb0.Ontheotherhand,N.FakhruddinhasgiventhefollowingtwoexamplesinwhichPsmooth\Sdisemptyforalld&#xr]TJ;&#x/F48;&#x 11.;镒&#x Tf ;.9; 2;.69; Td;&#x [00;0.Example5.14.LetXbetheimageofthe4-upleembeddingP1Z!P4Z.ThenXissmoothoverZ.Iff2Psmooth\Sdforsomed&#xr]TJ;&#x/F48;&#x 11.;镒&#x Tf ;.9; 2;.69; Td;&#x [00;0,thenHf\X'`SpecAiwhereeachAiistheringofintegersofanumbereldKiunramiedaboveallniteprimesofZ,suchthatP[Ki:Q]=4d.TheonlyabsolutelyunramiednumbereldisQ,soeachAiisZ,andHf\X'`4di=1SpecZ.Then4d=#(Hf\X)(F2)#X(F2)=#P1(F2)=3,acontradiction.Example5.15.LetXbetheimageofthe3-upleembeddingP2Z!P9Z.ThenXissmoothoverZ.Iff2Psmooth\Sdforsomed&#xr]TJ;&#x/F48;&#x 11.;镒&#x Tf ;.9; 2;.69; Td;&#x [00;0,thenHf\XisisomorphictoasmoothpropergeometricallyconnectedcurveinP2Zofdegree3d,henceofgenusatleast1,soitsJacobiancontradictsthemaintheoremof[Fon85]. 22BJORNPOONENDespitethesecounterexamples,P.Autissierhasprovedapositiveresultforaslightlydierentproblem.AnarithmeticvarietyofdimensionmisanintegralschemeXofdimensionmthatisprojectiveand atoverZ,suchthatXQisregular(ofdimensionm�1).IfOKistheringofintegersofaniteextensionKofQ,thenanarithmeticvarietyoverOKisanOK-schemeXsuchthatXisanarithmeticvarietyandwhosegenericberXKisgeometricallyirreducibleoverK.ThefollowingisapartofTheoreme3.2.3of[Aut01]:LetXbeanarithmeticvarietyoverOKofdimensionm3.ThenthereexistsaniteextensionLofKandaclosedsubschemeX0ofXOLsuchthat(1)ThesubschemeX0isanarithmeticvarietyoverOLofdimensionm�1.(2)WhenevertheberXpofXabovep2SpecOKissmooth,theberX0p0ofX0abovep0issmoothforallp02SpecOLlyingabovep.ActuallyAutissierprovesmore,thatonecanalsocontroltheheightofX0.(HeusesthetheoryofheightsdevelopedbyBost,Gillet,andSoule,generalizingArakelov'stheory.)ThemostsignicantdierencebetweenAutissier'sresultandthephenomenonexhibitedbyFakhruddin'sexamplesistheniteextensionofthebaseallowedintheformer.AcknowledgementsIthankErnieCrootforaconversationatanearlystageofthisresearch(in2000),DavidEisenbudforconversationsaboutSection5.5,andMattBaker,BrianConrad,andNaj-muddinFakhruddinforsomeothercomments.IthankPascalAutissierandOferGabberforsharingtheir(nowpublished)preprintswithme,andJean-PierreJouanolouforsharingsomeunpublishednotesaboutresultantsanddiscriminants.Finally,Ithanktherefereeforanextremelyhelpfulreport,whichincludedamongotherthingsthestatementandproofofLemma5.9.References[Aut01]PascalAutissier,PointsentiersettheoremesdeBertiniarithmetiques,Ann.Inst.Fourier(Grenoble)51(2001),no.6,1507{1523.[Aut02]PascalAutissier,Corrigendum:\IntegerpointsandarithmeticalBertinitheorems"(French),Ann.Inst.Fourier(Grenoble)52(2002),no.1,303{304.[Dwo60]BernardDwork,Ontherationalityofthezetafunctionofanalgebraicvariety,Amer.J.Math.82(1960),631{648.[EGAIV(4)]A.Grothendieck,Elementsdegeometriealgebrique.IV.Etudelocaledesschemasetdesmor-phismesdeschemasIV,Inst.HautesEtudesSci.Publ.Math.(1967),no.32,361.[Fon85]Jean-MarcFontaine,Iln'yapasdevarieteabeliennesurZ,Invent.Math.81(1985),no.3,515{538.[Ful84]WilliamFulton,Introductiontointersectiontheoryinalgebraicgeometry,PublishedfortheConferenceBoardoftheMathematicalSciences,Washington,DC,1984.[Gab01]O.Gabber,OnspacellingcurvesandAlbanesevarieties,Geom.Funct.Anal.11(2001),no.6,1192{1200.[GM91]FernandoGouv^eaandBarryMazur,Thesquare-freesieveandtherankofellipticcurves,J.Amer.Math.Soc.4(1991),no.1,1{23.[Gra98]AndrewGranville,ABCallowsustocountsquarefrees,Internat.Math.Res.Notices(1998),no.19,991{1009.[Har77]RobinHartshorne,Algebraicgeometry,Springer-Verlag,NewYork,1977,GraduateTextsinMathematics,No.52.[Hoo67]C.Hooley,Onthepowerfreevaluesofpolynomials,Mathematika14(1967),21{26.[Kat99]NicholasM.Katz,Spacellingcurvesoverniteelds,Math.Res.Lett.6(1999),no.5-6,613{624. BERTINITHEOREMSOVERFINITEFIELDS23[Kat01]NicholasM.Katz,Correctionsto:Spacellingcurvesoverniteelds,Math.Res.Lett.8(2001),no.5-6,689{691.[Kle86]StevenL.Kleiman,Tangencyandduality,Proceedingsofthe1984Vancouverconferenceinalgebraicgeometry(Providence,RI),CMSConf.Proc.,vol.6,Amer.Math.Soc.,1986,pp.163{225.[KS99]NicholasM.KatzandPeterSarnak,Randommatrices,Frobeniuseigenvalues,andmonodromy,AmericanMathematicalSociety,Providence,RI,1999.[LW54]SergeLangandAndreWeil,Numberofpointsofvarietiesinniteelds,Amer.J.Math.76(1954),819{827.[Poo00]BjornPoonen,VarietieswithoutextraautomorphismsIII:hypersurfaces,preprint,2May2000.[Poo03]BjornPoonen,Squarefreevaluesofmultivariablepolynomials,DukeMath.J.118(2003),no.2,353{373.[PS99]BjornPoonenandMichaelStoll,TheCassels-Tatepairingonpolarizedabelianvarieties,Ann.ofMath.(2)150(1999),no.3,1109{1149.[Ser65]Jean-PierreSerre,ZetaandLfunctions,ArithmeticalAlgebraicGeometry(Proc.Conf.PurdueUniv.,1963),Harper&Row,NewYork,1965,pp.82{92.[Wei49]AndreWeil,Numbersofsolutionsofequationsinniteelds,Bull.Amer.Math.Soc.55(1949),497{508.DepartmentofMathematics,UniversityofCalifornia,Berkeley,CA94720-3840,USAE-mailaddress:poonen@math.berkeley.edu