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HILBERT SCHEMES OF FAT r PLANES AND THE TRIVIALITY OF CHOW GROUPS OF COMPLETE INTERSECTIONS

22 pages
HILBERT SCHEMES OF FAT r-PLANES AND THE TRIVIALITY OF CHOW GROUPS OF COMPLETE INTERSECTIONS ANDRE HIRSCHOWITZ AND JAYA NN IYER Abstract. In this paper, we investigate the question of triviality of the rational Chow groups of complete intersections in projective spaces and obtain improved bounds for this triviality to hold. Along the way, we study the dimension and nonemptiness of some Hilbert schemes of fat r-planes contained in a complete intersection Y , generalizing well-known results on the Fano varieties of r-planes contained in Y . Contents 1. Introduction 1 2. Strong planes 5 3. Restricted flag-Hilbert schemes 7 4. Fat planes in complete intersections 9 5. Nonemptiness 14 6. Spannedness 18 7. Conclusion and future work 20 References 21 1. Introduction The aim of this paper is to investigate the triviality of the low-dimensional rational Chow groups for certain projective varieties. If Y is a nonsingular complete intersection of multidegree (d1, · · · , ds) in a projective space Pn, and n is sufficiently large with respect to the degrees, it is known that, for small values of r, the rational Chow group QCHr(Y ) := CHr(Y ) ? Q is trivial, namely one-dimensional (generated by the linear sections).

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HILBERTSCHEMESOFFATr-PLANESANDTHETRIVIALITYOFCHOWGROUPSOFCOMPLETEINTERSECTIONSANDRE´HIRSCHOWITZANDJAYANNIYERAbstract.Inthispaper,weinvestigatethequestionoftrivialityoftherationalChowgroupsofcompleteintersectionsinprojectivespacesandobtainimprovedboundsforthistrivialitytohold.Alongtheway,westudythedimensionandnonemptinessofsomeHilbertschemesoffatr-planescontainedinacompleteintersectionY,generalizingwell-knownresultsontheFanovarietiesofr-planescontainedinY.Contents1.Introduction2.Strongplanes3.Restrictedag-Hilbertschemes4.Fatplanesincompleteintersections5.Nonemptiness6.Spannedness7.ConclusionandfutureworkReferences1579418120121.IntroductionTheaimofthispaperistoinvestigatethetrivialityofthelow-dimensionalrationalChowgroupsforcertainprojectivevarieties.IfYisanonsingularcompleteintersectionofmultidegree(d1,∙∙∙,ds)inaprojectivespacePn,andnissufficientlylargewithrespecttothedegrees,itisknownthat,forsmallvaluesofr,therationalChowgroupQCHr(Y):=CHr(Y)Qistrivial,namelyone-dimensional(generatedbythelinearsections).ThepreciseconjecturalboundonthemultidegreesforthetrivialityfollowsfromthestudyoftheHodgetypeofthecomplementaryopenvarietyPnYinitiatedbyDeligne[De]andfollowedbyworksofDeligne-Dimca[De-Di]andEsnault-Nori-Srinivas0MathematicsClassificationNumber:14C25,14D07,14D22,14F40.0Keywords:Completeintersections,Chowgroups,Hilbertschemes.1
2A.HIRSCHOWITZANDJ.N.IYER[Es],[EsNS].Aformulationoftheconjecturedboundwasmadein[Pa,Conjecture1.9],whichsays:Conjecture1.1.SupposeYPnisasmoothcompleteintersectionofmultidegree(d1,∙∙∙,ds),andletrbeanonnegativeinteger.IfsXr(max1isdi)+din1=ithenQCHr(Y)istrivial.Thecaseof0-cycleshasbeenhandledbyRoitman[Ro].ConcerningageneralcubichypersurfaceYPn,C.Schoen[Sc]showedthetrivialityQCH1(Y)'Qwhenn7andParanjape[Pa]obtainedthesharpboundinthiscaseshowingthetrivialityof1-cycleswhenn6.Healsogavethefirstfiniteboundforgeneralcompleteintersections.Lateron,abetterboundwasobtainedbyEsnault,LevineandViehweg:Theorem1.2.[EsLV]WeconsideracompleteintersectionYinPnofmultidegree(d1,∙∙∙,ds),withd1...ds2andanintegerrsatisfyingXsdi+r.ni=1r+1Intheparticularcased1=...=ds=2,wesupposefurthermorer<s.ThenQCHr0(Y)'Q,foreachr0r.Intheremainingcasewhered1=...=ds=2andsr,thesameconclusionasaboveholdsassumingthemodifiedinequalitys(r+2)nr+s1.TheirinequalityisthesharpconditionfortheChowgroupof0-cyclesofthevarietyofr-planesinthegeneralcompleteintersectionYtobetrivial.Hence,astheyclaim,theirboundcannotbeimproved”byonlyconsideringrationalequivalencesofr-planesamongr-planes”.ForYanonsingularhypersurface,animprovedboundhasbeenobtainedbyJ.Lewis[Le2],A.Otwinowska[Ot].Ourcontributionrecoverstheirboundforhypersurfaces(relaxingthenonsingularityassumption)byacompletelydifferentmethod,andsomehowimprovestheboundof[EsLV]inthehigher-codimensionalcase,aswewillseebelow.Thecentralnotionsinourapproacharethoseoffatandstrongplanes,whichappearatleastimplicitlyin[EsLV],andgobacktoRoitmanforthe0-dimensionalcase.