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In this paper,We dealwithnormaltwo-dimensionalsingularities with multiplicity two・We callsuch a slngulanty double point. The purpose of this Faper is to give an estimate of the arithmeticgenusfbrdoublepolntSinterrnSOrHorikawa’scanonical resolutionヱInd a p‘J-fbrmlla fbr some elass ofdouble points・ltis known that,by uslng the data obtained n・Om the canonicalreso1ution,the geometric genus fbr double polntSis fbrmulated,and rationaldouble polntS and elliptic double polntS are Characterized. WeglVeaCharacterizationofdoublepolntSWiththearithmeticgenus two. §1.Introductiom Let(S,P)be agerm ofnormaltwo-dimensionalvariety over Cat apoint P・Wecal=tarlOrmaltwo-dimensionalsingularity・Letゆ:(S,A)→(S,P)bea resolution with the exceptionalset A・In his fhmous paper,M.Artin r2】 introduced animportant divisor,Whichis called fbndamentalcycle:Z= min(CJpositivecycleon鼠C・Ai≦Ofbranyi),WhereA=UAiisthedecomPOSition toirreducible components orthe except10nalset A,and showed that P。(Z)=Oirand onlyifRl¢*F云=0.Ne礼Wagreich[10]irltrOduced twonumericalinvariants fbr two-dimensionalslngulanty,Which are called geometric genus..Pf)FdimRIM-s,andarithmeticgenus:Pa=maX(pa(C)lCispositive CyCle on S),Wherepa(C)is thevirtualgenus ofC.Hefbundarelationofthe threeinvariants:0≦pu(Z)≦p。≦pg・We note that theseinvariants areindePendent ofresolution・He also studied the class ofe11iptlC Slnguiarlty,Whichis de且ned by p。=1,the condition ofwhichis equlValent to the condition that ReceivedJunell、ヱ002. RevisedJune30、ユ005. Masakazu TAKAMURA 36 pa(Z)=1.Sinceitis known that fundamentalcycleis obtained by using the computation sequence,We Can eaSily compute p。(Z)丘om the weighted dual graphoftheexcept10nalsetofaresolution・However,inthecaseofpa≧2,in general,Wehavep。(Z)<p。,andsinceitisverydifnculttocomputearithmetic genus,Ontheclassofsingularitywithmorelargearithmeticgenus,uSefultheory is not developed yet. In this paper,We discuss normaltwo-dimensionalslngularities with multiplicity two,Whichiscalled double polntS・We resoIvea double polnt byHorikawa’smethod[4,5,6]・Theresolutioniscalledcanonicalresolution・E・Horikawa (See[4],Lemma6)provedthatpg=(1/2)∑た0γi(yi+1),Wheretheinvariants)・i areeasily obtained byapplyingHorikawa’s canonicalresolution・Fromthis,We have the characterization of the rational double points as the class satisfying γi=Ofbralli.In[9],M・Tomarishowedcertainlowerestimateofpainterms Ofγ(・Heusedthistocharacterizetheclassofpa=lbyγi・However,ingeneral、 p〟is not a function ofγi・In this paper,We Can glVe an uPPer eStimate of the di触rence ofthe arithmetic genus and the virtualgenus ofa divisor which is easily computed,by the multiplicity of the branchlocus or double polnt (CorollaryCinSection4).Thisestimateisthebestasthelinearbound・Further Wegiveap。-fbrmulafbrsomeclassofdoublepoints(TheoremDinSection4), and acharacterizationofdoublepointswith p。=2intermSOfγi(Corollary E in Section4).We can負nd generalstudies on double pointsin[7,9]・ We explain the content ofeach section・ Section2isdevotedtoareview of-Horikawa’scanonicalresolutionofdouble points.Weremarkthatadoublepointiswrittenas((z2=g(x,y)),0)・Firstwe regard(z2=g(x,y))asasurfacecontainedinatriviallinebundleover(x,y)plane,Whichisthedoublecoveringwithbranchlocus(g(x,y)=0)inthebase SPaCe・We obtain a base space by blowlng-uP at an apPrOPrlate Slngularlty Of thebranchlocus,andaccordingto arule,Wede点ne aline bundle overthebase SPaCe・Wecandefine a normalsurface,Whichis double coverlng OVerthebase SPaCe,inthelinebundle,anddefine a birationalproper mapplng Ofthe surface OntO the orlglnalsurface.Ifthere exists a slngularlty Ofthebranchlocusofthe double covenng,We rePeatthisprocess.Afterfiniteprocesses,We Obtain anonslngularsurfacein aline bundle.Then thecompositionofthesemapplngSglVeS a resolution. InSection3,WeglVeSeVeralrepresentationsofthearithmeticgenusofdouble polntSbyuslngthecanonicalresolution・Roughlyspeaking,thearithmeticgenus is determined by theinfbrmation ofthe branchlocus.Let q be the associated COVerlngtranSfbmation oftheresolutionspace ofordertwo.We showthatthe On the estimate of the arithmetic genus 37 arithmeticgenusequalsthemaximumofthevirtualgenusofq-invariante恥ctive cycles,andwerepresentp。-Pa(Ys,P)bytheinvariants rOdd computed伽m γiand Vmincombinatoria11ydetermined什omcyclesonthebasespace(Theorem A)・Here Ys、Pisthecycleontheresolutionspace,Whichisdetermined丘omthe CanOnicaldivisor,and pLL(持、p)iscomputed丘omγi・ InSections4and5,uSlng therepresentation obtainedin Section3,We Will give a detailed estimate ofpu-P。(Ys,P)(Theorem B)・From Theorem B,We Obtainan estimatep〟-P〟(Yi,P)≦(1/8)ml,Wheremlisthemultiplicityofthe branchlocus at p(Coro11ary C).FurthermOre Wegive an example such that the equality holdsin the aboveinequality.In such a sense,We glVe the best estimate.Alowerestimate ofthe arithmeticgenusdue to M・Tomariiswritten asp‘J(Ys.p)≦p‘Jinourterminology(see【9],Lemmal)・Fromourupperestimate and Tomari’slower estimate,under the assumptlOn that ml≦8,We glVe a P(,-fbrmula:Pt,=Pu(センチs、P)(Theorem D)・Since wecaneasilycompute p,,(Ys、P) n・Omtherelation:PLJ(rT、′,)=∑フー,:eV。n)・f/4+∑)・l:。dd()・f-1)/4+l,itseemstobe useful.As a corollary ofTheorem D,We Obtain the characterization ofdouble POintswithp”=2(CorollaryE).Usingthischaracterization,WeCanClassi~the weighteddualgraphortheminimalresolutionofthedoublepointswithpEL=2[8]. Theessentialpart ofthe proofofTheorem Bis glVenin Section5・ Iam very graten11to Proftssor M.Tomarifbr theinspired discussion・ §2.Horikawa’s CanonicalResolution In this section,We Sha11review Horikawa’s canonicalresolution(see[4], Section2,[5】,Section3and【6=7.4)・Theresolutionisappliedfbrdoublepoints・ Let(SトPl)be a double point.We may assume that(Sl,Pl)is represented as fbllows: ∫1=((∫,γ,ニ)∈ダItニ2=♂(∫,J′))and 釣=(0几0) (See【ll).Further we may assume that there exists no singularity on SleXCePt pl.Here Flisatriviallinebundleoveran openneighborhood U10ftheorlgln plinC2((x,y)-Plane),andg:Ul→Cisareducedholomorphicfunctionwith g(0,0)=0,and=isthe点bercoordinateofFl・Let¢1bethenaturalprQjection OfS10ntO Uland Blbethebranchlocusig(x,y)=0)ofthedoublecovering Sl・Then the slngularities orthe surface SICOrreSPOnds to those ofthe branch locusβト From now on,We uSe the fbllowing notation.Fora realnumber a,回= max(n∈Zla≧n),andfbranintegraldivisorD,Wedenoteby[D]alinebundle associated to the divisor D. Masakazu TAKÅMURA 38 We constructthearst stepofthecanonicalresolution・Let7Tl:U2→Ulbe theblowlng-uPOfUlatPl.Wede6nealinebundleFiwiththebasespace U2 and a reduced divisor B20n U2aS fbllows: 蔦=ポ(ダ1)戯=ポ(β1)-2 HeremlisthemultiplicityofBlatPlandEtheexceptlOnalcurveof冗l・Then, 丘omtheisomorphism2Fl空[Bl],We Obtaintheisomorphism2Fゝと[B2]・ We represent the divisor B2bylocalequations g2k,Where g2kis a holomorphicftlnCtionontちkand(叛k)isanopencoveringofU2・Moreoverwemay assumethatFiisisomorphictoatriviallinebundleFk over U2k・Thenwecan de丘nethe double coverlng S2k OVer U2k aS fbllows: 裁た=‡(∫2ん,γ2尤,ニ2〟)∈鳥汗弦=g2バズ2尤,γ2た))・ Since2Fi空[B2],gluing S2k,We Obtain a surfhce S2in Fi,Whichis a double COVerlng OVer叛.Since Blhas no multiple component,B2has no multiple COmPOnent・lt fbllows that S2is a normalsurface・ We denote byゆ2the prqiection ofSionto U2・Then we can de触e the naturalbirationalholomorphic mapping元10fS20ntO SISuCh thatゆl。元1= 花l。¢2.In this way,We Obtain the quadruplet(B2,U2,S2,蔦)鉦0rnthe quadruplet(Bl,Ul,Sl,Fl). 1f B2has a slngularity,We Choose one of the slngularities or B2aS the centeroftheblowlng-uPfbrthesecondstep.SincetheslngularltyP2isadouble point,bythesamewayasabove,thequadruPlet(B3,U3,S3,蔦)isobtained丘om (βユ,伍,∫2,為). Afterweappliedtheaboveoperationoffinitetimes(rtimes),thebranchlocus B,+1becomestonon-SlngularandweobtainthefbllowlngCOmmutativediagram・ DIAGRAM 2.1.

On the estimate of the arithmetic genus for normal two-dimensional singularities on double coverings