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1、COMMENTSONVALUATIONSASSOCIATEDTOSYSTEMSOFVERTICES/EDGESANDTHEMAINTHEOREMOFPOP-STIXShinichiMochizukiUpdatedMay2,2011Letkbeanarbitrarycompletediscretevaluationfieldofmixedcharacteristicdefwhoseresiduecharacteristicwedenotebyp,kanalgebraicclosureofk,Gk=Gal(k/k),Σ
2、asetofprimesthatcontainsaprimel=p,Xaproperhyperboliccurveoverk.Suppose,further,thatkisl-cyclotomicallyfull,i.e.,thattheimage××ofthel-adiccyclotomiccharacterGk→ZlisopeninZl.Write(Σ)ΠXΠXforthegeometricallypro-Σquotientofthe´etalefundamentalgroupΠXofX.Thus,(Σ)
3、wehaveanaturalsurjectionΠXGk.Let...→Xi+1→Xi→...[whereirangesoverthepositiveintegers]beacofinalsystemoffinite´etaleconnected(Σ)GaloiscoveringsofXwithstablereductionarisingfromopensubgroupsofΠXand(Σ)s:Gk→ΠX(Σ)asectionofΠXGk.Theninthe“CommentsonaCombinatorialVer
4、sionoftheSectionConjectureandtheMainTheoremofPop-Stix”datedMarch3,2011(cf.[CbSC],(5)),weshowedthat(∗v/e)[afterpossiblypassingtoacofinalsubsystemofthegivensystemofcoverings]thereexistseithera[notnecessarilyunique]systemofvertices...vi+1vi...ora[notnecessaril
5、yunique]systemofedges...ei+1ei...—i.e.,eachvi(respectively,ei)isanirreduciblecomponent(respectively,node)ofthespecialfiberofthestablemodelXiofXithatisfixedbythenaturalactionoftheimageIm(s)ofthesections;theimageoftheTypesetbyAMS-TEX12SHINICHIMOCHIZUKIirreduci
6、blecomponentvi+1(respectively,nodeei+1)inXiiscontainedintheirreduciblecomponentvi(respectively,nodeei).Inthepresentnote,weverify(cf.(1),(2)below),bymeansofaquiteelementaryargumentinschemetheory/commutativealgebra,that(∗val)suchasystemofverticesoredgesdetermin
7、esasystemofvaluationsofthefunctionfieldsKioftheXithatarefixedbythenaturalactionofIm(s).Inparticular,weobtainaproofofthemaintheoremofPop-Stix(cf.[PS])bymeansofelementarygraph-theoreticandscheme-/ring-theoreticconsiderations,withoutresortingtotheuseofhighlynontri
8、vialarithmeticresultssuchasTamagawa’s“resolutionofnonsingularities”[i.e.,themainresultof[Tama]].Here,werecallthatthisresultof[Tama]depends,inanessentialway,onhighlyarithmeticargumentsthat
