Linearly normal curves with degenerate general hyperplane section

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1、HiroshimaMath.J.32(2002),217–228LinearlynormalcurveswithdegenerategeneralhyperplanesectionEdoardoBallico,NadiaChiarli,SilvioGrecoAbstract.Westudylinearlynormalprojectivecurveswithdegenerategeneralhyperplanesection,intermsofthe‘‘amountofdegeneracy’’ofit,givin

2、gacharac-terizationand/oradescriptionofsuchcurves.1.IntroductionWeworkoveranalgebraicallyclosedfieldKofarbitrarycharacteristic.By‘‘curve’’wealwaysmeanalocallyCohen-Macaulaypurelyone-dimensionalprojectivescheme.Recallthatanon-degeneratecurveYJPniscalledlinearl

3、ynormalif0n0thenaturalmapHðP;OPnð1ÞÞ!HðY;OYð1ÞÞisbijective,orequivalentlyifH1ðPn;Ið1ÞÞ¼0.YLinearlynormalcurvesoccurinaverynaturalsetting.IndeedifwestartwithanabstractprojectiveschemeYandalinebundleLAPicðYÞ,withLveryample,itisverynaturaltoconsiderthecompletee

4、mbeddingofYintheprojectivespacePðH0ðY;LÞÞinducedbyH0ðY;LÞ.Theaimofthepaperistogiveadescriptionand,wheneverpossible,tocharacterizelinearlynormalcurvesintermsofthe‘‘amountofdegeneracy’’ofthegeneralhyperplanesection.nWefixsomenotation.LetYJPbeanon-degeneratecurv

5、eandletC:¼Yred.Setd:¼degðYÞ,d:¼degðCÞands¼sðYÞ:¼dimðhYVHiÞ,whereHisageneralhyperplane.Ourfirstresultisthefollowingcharacterizationoflinearlynormalcurveswithmaximallydegenerategeneralhyperplanesection(i.e.s¼1).nTheoremA.LetYJPðnb3Þbeanon-degeneratecurve.Thenth

6、efollowingareequivalent:(i)s¼1andYislinearlynormal;(ii)degðYÞ¼2andpaðYÞ¼nþ2.2000MathematicsSubjectClassification.Primary14H50;Secondary14M99,14N05.Keywordsandphrases.linearlynormalcurve,unrerucedcurve,locallyCohen—Macaulaycurve,projectivescheme,hyperplanesec

7、tion,zero-dimensionalscheme.SupportedbyMURSTandGNSAGA-INDAMintheframeworkoftheproject‘‘Problemidiclassificazioneperschemiproiettivididimensionepiccola’’.218EdoardoBallicoetal.AcompleteclassificationandadescriptionofsuchcurvesisgiveninExample5.1.5Ournexttheorem

8、dealswithcurvesinP.RecallthatifYJP5isanon-degeneratecurve,withdegenerategeneralhyperplanesectionhavingCirreducible,notaline,thenhCiisaplaneands¼3(see[1],Th.2.1and[2],Th.2.5).5TheoremB.LetYJPbeal