ContentsIntroductioni01DimensionofFanovarietiesi02Thelowdegreelimitii - PDF document

1 ContentsIntroduction.i0.1.DimensionofFan
ContentsIntroduction.i0.1.DimensionofFanovarietiesi0.2.Thelow-degreelimitii0.3.Applicationtounirationalityinlowdegreeiii0.4.Furtherworkiii0.5.Incidencecorrespondencesiv1.PlanesonHypersurfacesinGeneral.11.1.DenitionoftheFanovariety11.2.EstimatingthedimensionofFk(X)21.3.Proofsbyexample.51.4.Fk(X)hastheestimateddimensionforgeneralX72.FanoVarietiesintheLow-DegreeLimit.112.1.Notationandterminology112.2.Arstresult112.3.Constructions132.4.Explicitdescriptionofthebersoftheresidualvarieties172.5.Numbers192.6.Low-degreesmoothhypersurfacesdonothavetoomanyk-planes203.UnirationalityofSmoothHypersurfacesofLowDegree.273.1.Preparatoryresults,andcombs283.2.Statementofresultsconcerningunirationality303.3.Constructions313.4.Proofofunirationalityofsmoothlow-degreehypersurfaces33Acknowledgements.36References37 iiAUTHOR:ALEXWALDRONADVISOR:JOEHARRIStobethenumberofconditionscuttingoutFk(X):HoweverFk(X)isasubvarietyoftheGrassmannianG(k;n);andthereforesuchanargumentfailstoshowthatFk(X)isinfactnon-emptyif0:Indeed,therequirementd3isessential,asshowninRemark1.8.And,althoughtheargumentwillbeentirelyclassical|adimensionco

2 untviaincidencecorrespondences(two,inthi
untviaincidencecorrespondences(two,inthiscase)|theclaimwasestablishedonlythroughtheresultofHochsterandLaksov[1]in1987.Interestingly,inthecase(n;d;k)0;theproofthatFk(X)isnon-emptyforallXPnofagivendegreedwilldependonprovingtheexistenceofahypersurfaceX0suchthatFk(X0)hasdimensionexactly(n;d;k)(seeRemark1.9).Wewillgiveseveralsuchexamples,therebyprovingparticularcasesoftheclaim.Forarbitraryn;d;k;wewillprovethatageneralsurfacehasFanovarietyofdimension;evenwhileweareleftwithnowaytoexhibitsuchahypersurface,norwithmeanstocheckthataparticularhypersurfaceis\general"inoursense.Thisisthepowerofusing\incidencecorrespondences"(section1.2.1)inconjunctionwiththeTheoremonFiberDimension(Proposition1.3),incontrasttothenaivecomputationalargumentsuggestedattheoutset.0.2.Thelow-degreelimit.Thislastresultestablishesthatallhypersurfaceshave\enough"k-planescorrespondingtotheirdegreeanddimension,andthatageneralhyper-surfacehastheexpected-dimensionalfamilyofk-planes.Butthenextquestionremainsinscrutable:whichhypersurfaceshave\toomany"k-planes,bywhichwemeanaFanovarietyofdimensiongreaterthan?Weproposetwocriteriaforremedyin

3 gthissitua-tion,i.e.specifyingwhichhyper
gthissitua-tion,i.e.specifyingwhichhypersurfacesare\general"intheprevioussense.Therst,smoothness,isnaturalespeciallywhenworkingoveraeldKofcharacteristiczero(aswewillinCh.2and3).However,itisnotsucient.Inthecasek=1oflines,thecanonicalexampleofasmoothhypersurfacefails:ifchar(K)=0;theFanovarietyoftheFermathypersurfaceofdegreed=n+1inPnhasdimensionn�3(see[5]),greaterthantheestimateddimension(n;d;1)=2n�3�d=n�4:Second,weworkinthelimitoflowdegreecomparedtodimension.Thisisarealmofbroadinterestpertainingtoseveraldierentelds:seeforexampleKollar'sarticle[12].Byitself,however,lowdegreeisclearlyinsucientforFk(X)tohavetheexpecteddimension,asseenbyconsideringanyreduciblehypersurface.Workinginthelow-degreelimitisacommonapproachfor\specializing"aknownfactaboutgeneralhypersurfacestosmoothones.2Inthecaseoflines,k=1;thewell-knownDebarre-DeJongconjecture[3]assertsthatifdnandXissmooth,thendim(Fk(X))=(n;d;1)=2n�3�d;i.e.Xdoesnothave\toomany"lines.Bytheexamplejustgiven,thebounddnissharp,ifitholds.Thisconjectureisactivelypursued(see[3],[4]):theresulthasbeenestablishedford6[5],

4 andforseveralmonthsin2007aproofofthegene
andforseveralmonthsin2007aproofofthegeneralcasewasthoughttohavebeenfound.Therehasalsobeenprogressintheareaofrationalcurveslyingonhypersurfacesoflowdegree:StarrandHarris[10]showthatford(n+1)=2;ageneraldegreedhypersurfacecontainstheexpected-dimensionalvarietyofrationalcurvesofeachdegree.Ourassertionisthatforaxedk;ifXPnissmoothofdegreedn;thenXdoesnothavetoomanyk-planes.ThisistheresultofHarris,MazurandPandharipande[2]in1998,inwhichthesesameFanovarietiesarealsoshowntobeirreducible.Theproof 2Theterm\specialization"canrefertoamuchmoreinvolvedsetoftechniqueshavingthisfunction. ivAUTHOR:ALEXWALDRONADVISOR:JOEHARRISTable1.Themeaningofnd: d N0(d;k) U(d) k=1k=2k=3


2 469


0 3 52117250 3 4 342763662723391294 179124155 5 101710211025


10145 6 10821010310122 108790 ... ......... ... SetM(d;k):=�k+d�1d�1
+k�1:(ThisnumberwillreappearinLemma2.11,forthereasonthatifnM(d;k)thenXissweptoutbyk-planes.)In[9],itisshownusingmuchmoresophisticatedtechniquesthatifnM(d;k)+1andXissmooth,thenFk(X)hasatleastacomponentoftheexpecteddimension.ItisalsoshowntherethatifnM(d;k)t

5 henthegeneralk-planesectionofXisageneral
henthegeneralk-planesectionofXisageneralhypersurfaceinPk;whereasin[2]thissamefactwasprovedonlyforn�N0(d�1;k):So[9]hasmanagedtoimproveoneofthemainresultsofoursource[2]toapracticallevel,leavingoneoptimisticaboutsuchprospectsfortheresultswhichwedemonstratehere.Chen[11]hasalsoshownasacorollaryto[2]thattheFanovarietyisitselfunirationalinthelow-degreelimit.0.5.Incidencecorrespondences.Perhapsthemainideaofthispaperistotakean\incidencecorrespondence"(seesection1.2.1)betweentwofamiliesinordertogaininfor-mationaboutonebymeansoftheother(oftenusingtheTheoremonFiberDimension(1.3)).Anapologyisevennecessaryfortherepeateduseofthisideainallthreechapters|eventheunirationallyparametrizingvarietywillbebuiltusingthe\relativeFanovariety,"whichislocallyjustaformoftheincidencecorrespondence\I"ofChapter1!Moreover,anideasimilartothatoftherstchapterisappliedintheinductivestepofourproofofunirationality:inProposition3.12ofChapter3,wewillprovesurjectivitybyshowingthatthegenericberofamaphasappropriatelylowdimension,basedonthefactfromCh.2thatsmoothhypersurfacesoflowdegreedonotcontain\toomany"k-planes.(AlthoughinCh.1thisargumentismad

6 efortheprojectionontotheoppositefactor.)
efortheprojectionontotheoppositefactor.)Theubiquityofthisconstructionbecomesimpressive. 2AUTHOR:ALEXWALDRONADVISOR:JOEHARRISThusinordertoshowthatFk(X)isaclosedsubvarietyofG(k;n),itissucienttoshowthatFk(X)\Uisananevariety,foranysuchanepatchUG(k;n).Wlog,letXbeahypersurfacedenedbyasinglehomogeneouspolynomialf(X0;:::;Xn)ofdegreed;thegeneralcasewillfollowjustbytakingintersections.ForXahypersurface,Xitherestrictionofftovanishesidentically.Let2U;andparametrizethelinearspaceby[u0;:::;uk]7![Puii];withithei'throwofthematrixMfrom(2)(i.e.basisvectorfor).Wethensimplypluginthisparametrizationtoobtainapolynomialf(Puii)ofthesamehomogeneousdegreedintheparametersfuigofthespace.Thecoecientsaretheninhomogeneouspolynomialsintheaij;theirvanishingisnecessaryandsucientforftovanishidenticallyon,i.e.forthespacetobecontainedinthelocusff=0g=X.Thuswehaveobtainedpolynomialequationsintheaijdeningf2UjXg=Fk(X)\U;foranysuchU:HenceFk(X)isaclosedsubvarietyofG(k;n);andaprojectivevarietyitself.NotethatifXisquasi-projective,withprojectiveclosure X;th

7 entheFanoVarietyFk(X)Fk( X)isalocal
entheFanoVarietyFk(X)Fk( X)isalocallyclosedsubvariety.Thisissimplybecauseforanytwofamiliesofsubvarietiesofavariety,thesubsetofdisjointpairsisopen([13],Ch.4).Remark1.2.EliminationtheorycanbeusedoneachelementofanitecoverofpatchesfUi=A(k+1)(n�k)g;usingtheexplicitpolynomialdescriptionofFk(X)\Uijustgiven,toanswertheopeningquestionofwhetherornotagivenvarietyXactuallycontainsak-plane.1.2.EstimatingthedimensionofFk(X).1.2.1.Incidencecorrespondence.InordertondanestimateofthedimensionofFk(X)forahypersurface,wesetupanincidencecorrespondencebetweenhypersurfacesandk-planes.LetPNbetheprojectivespaceofhomogeneouspolynomialsofdegreedinn+1variables,soN=�n+dd
�1:ThenletI=f(f;)2PNG(k;n)jf()=0g:GivenanytwofamiliesofprojectivevarietiesinPn;thesetofpairssuchthattherstsubvarietyiscontainedinthesecondisaconstructibleset(see[13]).Here,IisinfactaclosedsubvarietyofPNG(k;n);whichcanbeshownalongmuchthesamelinesastheaboveargumentforFk(X).ChooseananecoordinatepatchU0ofPNG(k;n).UndertheSegreembedding,U0=U1U2fortwoanecoordinatepatchesU1PNandU2G(k;n).ThenU1isoftheformU1=f[f(

8 T)]2PNjf(T)=Th0 +Xh 6=h0 eh Th g;andwele
T)]2PNjf(T)=Th0 +Xh 6=h0 eh Th g;andweletU2G(k;n)haveanecoordinatesaijcomingfromthematrixMof(2).ThenviatheSegreembedding,oneobtainsinhomogeneouscoordinatesonU0=U1U2aspairs(feh g;faijg):Meanwhile,thepointsf2U1arestillhomogeneouspolynomials,andthepoints2U2arestillprojectivelinearspaces.Sowecanparametrizeasbeforeby[u0;:::;un]7![Pu``].Thenf()isahomogeneouspolynomialintheparametersu`,whosecoecientsaredoublyinhomogeneouspolynomialsinthecoecientseh offandtheentriesaijofthe 4AUTHOR:ALEXWALDRONADVISOR:JOEHARRISTheresultsofthischapterwillcomefromapplyingthesestatementstothemap1aswellasinseveralsimilarinstances.First,weapplyProposition1.3tothemap2tocomputethedimensionofI:Given2G(k;n)considerthesurjectivelinearmapfpolynomialsofdegreedonPng�!fpolynomialsofdegreedon=Pkggivenbyrestrictionto.Thekernelofthismapisalinearsubspaceofdimension�n+dd
��k+dd
;whoseprojectivizationisthetheberof2over:Thereforethebersof2areallirreducibleofthesamedimension,soIisirreducible(byTheorem11.14of[13]).FromProposition1.3,dim(I)=dim(G(k;n))+dim(&#

9 25;�12())=(k+1)(n�k)+n+dd
25;�12())=(k+1)(n�k)+n+dd�k+dd�1:Themap1cannotbedealtwithsostraightforwardly.However,itwillbeshowninsection1.4thatitsimagehasmaximaldimension(unlessd=2).Thismeansthat1isinjectivewhendim(I)N;and1issurjectivewhendim(I)N:Onceweestablishthis,wewillbeabletocountthedimensionbyapplyingProposition1.3.Untilthislastfactwasproven,by[1]in1987,thefollowingwasmerelya\dimensionestimate:"Theorem1.6.Fornk0andd3,let(n;d;k)=dim(I)�dim(PN)=(k+1)(n�k)�k+dd:(a)For(n;d;k)0;thesubvarietyofPNofhypersurfacesthatcontainak-planehascodimension�:(b)For(n;d;k)=0;everyhypersurfaceofdegreedinPncontainsak-plane,andageneralhypersurfacecontainsapositivenumberofk-planes.(c)For(n;d;k)&#x]TJ/;༕ ;.9; T; 12;&#x.479;&#x 0 T; [00;0;ahypersurfaceXhasdim(Fk(X))(n;d;k);withequalityforgeneralX.Bytheprecedingdiscussion,thistheoremwillbeprovedifwecanestablishthattheimageof1:I!PNhasmaximaldimension,i.e.issurjectivefor0andinjectivefor0:Atpresent,however,wecanproveonlysmallpiecesofthistheorem.It

10 isclearthatfor0;thevarietyofhypersu
isclearthatfor0;thevarietyofhypersurfacesthatdocontainak-planehascodimensionatleast�=dim(PN)�dim(I);sothefollowingisalreadyevident:Corollary1.7.For0;ageneralhypersurfacecontainsnok-planes.Alsonotethatfor&#x]TJ/;༕ ;.9; T; 21;&#x.046;&#x 0 T; [00;0;ifahypersurfaceXcontainsak-plane(i.e.liesintheimage1(I))thenitcontainsinnitelymanyk-planes,sincedim(Fk(X))=dim(�11(X))(n;d;k)&#x]TJ/;༕ ;.9; T; 21;&#x.046;&#x 0 T; [00;0:Thenontrivialtaskistoshowthatahypersurfacedoesinfactcontainak-plane:Example1.8.ThegeneralquadricthreefoldinP4containsno2-planes|thisisthecasen=4;k=2;d=2and(4;2;2)=0:For,inthefamilyofallquadrichypersurfaces,thesub-familyofconesoverquadricsurfaceshascodimension1;andeachsuchconecontainsatleasta1-dimensionalfamilyof2-planes.Thereforethissub-familymustequaltheimage1(I):Thestipulationd3isthusnecessaryin(b),andassuchitisnotobviousthat(b)holdsatall. 6AUTHOR:ALEXWALDRONADVISOR:JOEHARRIS1.3.2.Thecasek=2andabove.Wegivesimilarexamplesinthecasesk=2;d=4;5and=0:Fromtheseexampleswecaninductivelydescribepolynomial

11 sasrequiredforalld1;2mod3;provingal
sasrequiredforalld1;2mod3;provingallcasesk=2and=0:Example1.12.4(a)(n=7;d=4;k=2)ThequartichypersurfaceinP7=f[Z0;Z1;Z2;W0;W1;W2;W3;W4]gdenedbytheequationW0Z30+W1Z31+W2Z32+W3Z0Z1Z2+W4(Z0Z21+Z1Z22+Z2Z20)=0;containsthe2-planefWi=0gasanisolatedpointofitsFanovarietyof2-planes.(b)(n=9;d=5;k=2)ThequintichypersurfaceinP9=f[Z0;Z1;Z2;W0;:::;W6]gdenedbytheequationXi;j;kevenW`Zi0Zj1Zk2+W6(Z20Z1Z2+Z0Z21Z2+Z0Z1Z22)=0containsthe2-planefWi=0gasanisolatedpointofitsFanovarietyof2-planes.(c)(n=8;d=3;k=3)ThecubichypersurfaceinP8=f[Z0;:::;Z3;W0;:::;W4]gdenedbytheequation3Xi=0WiZ2i+W4(Z0Z1+Z1Z2+Z2Z3+Z3Z0)=0containsthe3-planefWi=0gasanisolatedpointofitsFanovarietyof3-planes.Toprovetheseassertions,onemustshowthatuponsubstitutingWi=PaijZjtheresultingformsspanK[Z]dasaijvaries.Then,since(n;d;k)=0;theyarealsolinearlyindependent,meaningthat=(aij=08i;j)istheonlyk-planecontainedintheanepatchU\Fk(X):Thisisseeninpart(a)bychoosingvaluesofthecoecientsaijasfollows:settingWi=Ziforeachi=0;1;2individually,weseethatallofthemonomialsZ4iareincludedinthespan.SettingW3=Ziforeachiindividually,wegeteachmonomialinvolvingallof

12 theZk:Now,settingW4=Zi;theonlymonomialno
theZk:Now,settingW4=Zi;theonlymonomialnotdivisiblebyanyZ3korinvolvingallthreeZiisZ2iZ2i+1;andtheseareexactlytheremainingmonomials.SowehaveshownthatallthemonomialsofK[Z]4areinthespan.Parts(b)and(c)aretreatedsimilarly.Forthecasek=2;onecandescribeinductivelyasetofpolynomialsdeninghyper-surfaceswiththerequiredproperty.Wehavetreatedthecasesd=4;5individuallyinExample1.12(a),(b),respectively.Then,givenFd(Z;W)ofdegreedasrequired,deneFd+3(Z;W;W0)=Z0Z1Z2Fd(Z;W)+W0aZd+20+W0bZd+21+W0cZd+22+dXk=1W0k(Zk+10Zd+1�k1+Zk+11Zd+1�k2+Zk+12Zd+1�k0): 4Part(a)herewasfoundbywritingoutthetheresultofsubstitutingWi=PaijZjintoanarbitrarypolynomial,andchoosingitscoecientssothattheresultspansK[Z]4:Part(b)wasfoundbywritingoutthedegreedmonomialsina2-simplexwiththedegreed�1monomialsinterlaced,andchoosingapolynomialthatspansK[Z]5undercertainchoicesofcoecientsaij:Part(c)wasfoundsimilarly. 8AUTHOR:ALEXWALDRONADVISOR:JOEHARRISi.e.dim(V)=dim(B)�1,andagenericm-tuple(Fi)Bwillsatisfytheconclusionofthelemma.Letp=[(aij)]2Ptberepresentedbytheranktmatrix(aij);andletF=�12(p)betheberof2overp:Wewillrstshowth

13 atthisberisirreducibleandhasdimensi
atthisberisirreducibleandhasdimension(4)dim(F)=mN(r;d�1)�N(r;d)+N(r�t;d):SinceitsuniqueP-coordinateisp,thevarietyFmaybeidentiedwithitsprojectionontoB;i.e.thesetofvectorsb=(F1;:::;Fm)2(K[x]d�1)msuchthatb(aij)~x=0.SolettingLbethecolumnvectoroflinearforms(aij)~x=L=(L1;:::;Lm)t;wethinkofFastheaneconefb2Bjb
L=0g:SoFissimplyalinearsubspaceofthevectorspaceB;henceirreducible.LetJk[x]betheidealgeneratedinL1;:::;Lm:Thenwehavetheshortexactsequenceofgradedrings(K[x](�1))m
L!K[x]!K[x]=J:SinceF=Ker(
L:(K[x](�1))m!K[x]);thisinducestheshortexactsequenceofvectorspaces0!F!(K[x]d�1)mL!K[x]d!(K[x]=J)d!0:SinceK[x]=Jisisomorphictoapolynomialringinr�tvariables,thissequencedirectlyyieldsthedesireddimensioncountfordim(F)in(4).Now,noticethatthemap2issurjectiveontoP,since(0;p)2V8p2P:Furthermore,dim(Pt)=mr�1�(m�t)(r�t)=N(r;d)�`�1�(m�t)(r�t)by[13]Prop.12.2andthegiven.So,sincetheberFwasarbitrary,fromCorollary1.4totheTheoremonFiberDimensionwehavedim(Vt)=dim(F)+dim(Pt)=mN(r;d�1)�N(r;d)+N(r�t;d)+N(r;d)�`�1�(m�t)(r�t)=(mN(r;d�1)�1)+N(r&

14 #0;t;d)�(m�t)(r�t)�`:Toshow3
#0;t;d)�(m�t)(r�t)�`:Toshow3,wenowmustprovethat(m�t)(r�t)+`N(r�t;d)foralltr;withourchoiceofmandr:Proceedbyinductionont:thebasecaset=0followsfromtheassumptionN(r;d)=mr+`:Nowassume(m�(t�1))(r�(t�1))+`N(r�(t�1);d):(m�t)(r�t)=r�t r�t+d(m�t)(r�t+d)r�t r�t+d(m�t+1)(r�t+1)sincemrtandd�12r�t r�t+d(N(r�t+1;d)�`)byinductionN(r�t;d)�`:Wehaveestablishedthatforalltr;dim(Vt)dim(Vr)=mN(r;d�1)�1:SinceV=SVt;wegetdim(V)=maxt(dim(Vt))=dim(Vr)=mN(r;d�1)�1;asdesired. 10AUTHOR:ALEXWALDRONADVISOR:JOEHARRISissurjective,ford3:5Hencethekernelofthismaphasdimensionrm�N(r;d)=(n;d;k)inourset-up,whichisexactlythestatementthatFj(Z)0foraijinalinearsubspaceofdimension(n;d;k):ThusFk(X0)\UU=Armisalinearsubspaceofdimension;whichisthereforethelocaldimensionofFk(X0)onU:Thisestablishespart(c)ofthetheorem.Remark1.14.Itissomewhatsurprisingthatourargumentestablishessurjectivityonlyinthecase(n;d;k)=0;andnotforthecasethatthedimensionofIisstrictlybiggerthanthato

15 fthetargetPN:However,thisisnottheonlyfai
fthetargetPN:However,thisisnottheonlyfailureofitskind,e.g.asurjectivemapofsheavesisnotingeneralsurjectiveonglobalsectionsunlessitisalsoinjective. 5Therequirementd3isnotincludedin[1].Theerrorissimplytheunjustiedclaimthattheinequalityatthebottomofp.237holdsfor\d=1"(whichisd�1=1inthispaper).Thesubsequentcombinatorialproofoftheinequalityford2iscorrect. 12AUTHOR:ALEXWALDRONADVISOR:JOEHARRISProposition2.3.Forn3;asmoothcubichypersurfaceXPnKisunirational,providedKisalgebraicallyclosedandchar(K)=0:Proof.Certainnotionsusedhere(suchasGrassmannbundles)willbemaderigorousinSection2.3immediatelyfollowingthisproof,whichwillthusbesomewhatcasual.FromTheorem1.6,wemaychooseaplane�ofsomedimensionl�0lyingonX(aline,ifoneprefers).Thenthevarietyof(l+1)-planescontainingthel-plane�isparametrizedbyPn�l�1=Pn=�:SinceXissmooth,ageneralsuch(l+1)-plane�meetsXinatheunionoftheplane�anda\residual"quadrichypersurfaceofdimensionl;callitQ:So,Xisbirationaltothetotalspaceofa\quadricbundle"|i.e.aPn�l�1-schemewhosebersarequadrics|overanopensubsetofPn�l�1: Tobeprecise,thequadrics

16 Qarethegenericbersoftheblow-up&
Qarethegenericbersoftheblow-up�:~X=Bl�(X)�!Pn�l�1;whichweviewasaquadricbundleoverPn�l�1viatheprojection�from�=Pn�l�1:ThepropertransformBl�()ofan(l+1)-plane�isagainan(l+1)-plane,andQ=Bl�()\~X:(Intherestofthisargumentwewillsometimesfailtomentionwhenapropertyholdsonlyoveranopen,densesubset.)Wewillshowthat~Xisunirationalbyviewingitasaquadricbundle.EachirreduciblequadricQisrationalindividually:projectingfromanypointofQtoahyperplaneofis1-1anddominant,thereforebirationalsincechar(K)=0:Wehopetopiecetogethertheserationalparametrizationsconsistently,inordertoparametrizeanopensubsetof~X:Thekeytodoingthisistondarationalsectionofthequadricbundle~X=fQj2Pn�l�1g;i.e.arationalmap:Pn�l�199K~Xwith()2Q:Ifwecouldndsuchasection;thenwewouldhavearationalmap(5)q7! ();q2G(1;);denedforeach:Thesemapsarenowclearlycompatibleover~X�E:Now,thespacesG(1;)formthe\Grassmannbundleoflines"(whichwewillsoonrefertoas\Grass1")intheprojectivebundleof(l+1)-planes;parametrize

17 dbyPn�l�1:AnyGrassmannbundleoverar
dbyPn�l�1:AnyGrassmannbundleoverarationalbaseisrational(seeProposition3.5,tofollow).Thisrationalmapfrom~XintotheGrassmannbundleofG(1;)'sis1-1anddominantovereach;henceisbirationaloverall.Inshort,thisparagraphhasshown: FANOVARIETIESANDUNIRATIONALITY13Lemma2.4.Thetotalspaceofafamilyofpointedquadricsoverarationalbaseisratio-nal.However,thereisnoguaranteeofndingsuchasection;i.e.achoiceofpointq2Qvaryingregularlywith2Pn�l�1:Saywetakel=1andlookonlyforpointsofQthatlieinsidetheline�:theningeneralZ:=Q\�willconsistoftwopoints,andwewillhavenowaytopickoneconsistently.Thusweareforcedtoabandonhopeofarationalparametrization,whichis1-1,andinsteadhopeforaunirationalparametrization,i.e.adominantmapPN99KX|or,equivalently,adominantmapfromarationalvarietytoX:(Althoughinfact,smoothcubicsurfacesinP3arerational.)Now,byanalogywehopetonda\unirationalsection"ofthebundleofthequadricbundle~X=fQg,i.e.amapfromarationalvarietyto~Xthathitsanopensubsetof2Pn�l�1:Fortunately,wewillndsucharationalvarietybydirectlyxingtheproblemofthepreviousparagraph(usingafamiliarconstruc

18 tion):considertheincidencecorrespondence
tion):considertheincidencecorrespondence =f(;p)jp2Z=Q\�gPn�l�1�:Thisvariety manifestlyhasa\unirationalsection"to~X;givenby(;p)7!p2Q:And isitselfarationalvariety,whichwewillproveshortlyinLemma2.5.Whatgoodisa\unirationalsection?"Itfurnishesarationalsectionnotof~Xbutofaquadricbundlethatdominates~X;namelythepullbackH:=~XPn�l�1 :ThisvarietyHisaquadricbundleover ;whoseberover(;p)isbydenitionthepointedquadric(p2Q):ThusbyLemma2.4,Hisrationalif isrational.WecandescribeHmoreconcretelyasfollows:H=f(q;;p)jq2Q;p2Zg~XPn�l�1�:SoHclearlydominates~Xand,pendingthefollowingresult,wehaveobtainedaunirationalparametrizationof~XandhenceofX:Lemma2.5.Theincidencecorrespondence isrational.Proof.Thevariety Pn�l�1�is(overanopensubset)abundleof(l�1)-dimensionalquadrichypersurfacesZ=Y\��overPn�l�1;soitseemsatrstthatwehavemadenoprogresstowardsaproof.However,consider asasabundleoveritsotherfactor,�:observethatapointp2�simplyimposesalinearconditionon2Pn�l�1;whichcorrespondstothetangenth

19 yperplaneTpXtoXatp:Infact, =fZg2
yperplaneTpXtoXatp:Infact, =fZg2Pn�l�1isinfactalinearsystemofhypersurfacesin�=Pl;parametrizedby2Pn�l�1:(ThiswillbeshownexplicitlyinSection2.4.)Furthermore,thissystemhasnobasepoints,sinceabasepointwouldbeasingularpointofX(seealso2.4).Thereforeeachberoftheprojection !�isahyperplaneinPn�l�1;so isaPn�l�2-bundleover�=Pl(byProposition3.3tofollow),hencerational.2.3.Constructions.Weproceedtointroduceseveralobjectsofwhichwewillmakeextendeduseinthischapterandthenext.ForthedurationofSection2.3,weletE!Bbeavectorbundleofrankr+1,withBandEirreduciblevarieties(i.e.integralquasi-projectiveK-schemes,inourterminology). FANOVARIETIESANDUNIRATIONALITY15OP(d)(d�0)overanopensubsetB1BsuchthatEjB1=B1VarehomogeneousformsofdegreedonVwithcoecientsregularoverB1:ThuswemayconcludethatOP(d)(PB1)=SymdE(B1);andtherefore(OP(d))=SymdE:AglobalsectionofOP(d)denesaclosedsubschemeofP!B:6LetSPVVbetheuniversallinebundleoverPV=G(1;V)|foreach`2PV;ShasberS`=`V:DeneO(�1)=OPr(�1)asthesheafofsectionsofS;anddenethe\tautologi

20 calsheaf"O(1)asthesheafofsectionsofS
calsheaf"O(1)asthesheafofsectionsofS(see[15]vol.2,ch.VIforacompletedescription).WemayalsodeneO(d)=O(1) d:Similarly,deneSP!PtobetheuniversallinebundleoverP=Grass0(E);i.e.thesub-bundleofPBEwhichhasbers(SP)(b;`b)=`bEb:WecannowdeneOP(�1)tobethesheafofsectionsofSP;OP(1)tobethesheafofsectionsofSP;andOP(d)=OP(1) d:Now,letB0BbeananeopensubsetsuchthatEjB0=B0VandPjB0=B0PV:Weemulatetheargumentof[15]vol.2,VI.1.4,Example2,tondthesectionsofOP(1)overB0PV:OnecanseefromthedenitionsthatSPjPB0=B0PVPVS;whichimpliesSPjPB0=B0PVPVS:ChoosecoordinatesfXgonPV;andconsidertheanecoordinatepatchesU=fX6=0gPV:Exactlyasin[15],thelinebundleSP!PVistrivialovereachopensetB0U;withtransitionfunctionsc=X=Xbetweenthem.SectionsoverB0Uareregularfunctions:theseareoftheform'=P=XdwithPahomogeneousformofdegreedintheXwithregularfunctionsonB0ascoecients.ForsuchasectiontoberegularonB0Uforall;wemusthavec'=XP=(XXd)regularonU

21 ;whichimpliesd=1:ThusthesectionsofS
;whichimpliesd=1:ThusthesectionsofSPoverPB0=B0PV;i.e.theelementsofOP(1)(B0PV);arehomogeneouslinearformswithregularfunctionsonB0ascoecients.Thisdescriptionthenclearlyextendstonon-aneopensubsetsB1BoverwhichPistrivial.Likewise,thesectionsofOP(d)overB1areformsofdegreed:ThesehomogeneouslinearformstakevaluesonE;sowemayconcludethat(OP(d))=SymdEford�0:Cartierdivisors.WewillthinkofaCartierdivisorDonaschemeXasaglobalsectionofthesheafMX=OX(denedin[17]II.6),whichagreesexactlywiththenotionofa\locallyprincipaldivisor"onanirreduciblevarietyintheclassicalsetting.Withthisdenition,aCartierdivisorDiseectiveiitcanberepresentedonanopencoverfUgbyelementsofOX(U)MX(U);called\localequations"forD.AneectiveCartierdivisorDthusdenesalocallyprincipalidealsheafIDandhenceaclosedsubschemeofX;referredtobythesameletterD:In[15]vol.2,VI.1.4,classesofCartierdivisorsareshowntobeequivalenttoinvertiblesheavesandlinebundles,whichallowsusthepushforwardandpullbackoperations,respectively(thoughonemayloselocaltrivialityinapplyingthepushforward).TheCartierdiv

22 isorsmanifestlyformagroup.Inthecaseofthe
isorsmanifestlyformagroup.InthecaseoftheprojectivebundleP=P(E);onecanseefromthepreviousdescriptionthatthenonzeroglobalsectionsofOP(d);foralld0;formagradedsubmonoidoftheeectiveCartierdivisors.Furthermore,if2H0(P;OP(d))and2H0(P;OP(d0))denesubschemesPPP;respectively,then=willbeaglobalsectionofOP(d0�d): 6NotethatthevectorbundlesOP(d);d�0;arenotnecessarilysub-bundlesoftrivialbundles,orevenprojectivelyembeddable;butthiswillremaintrueofallprojectivebundleswewillconsider,therebynotviolatingRemark2.2. FANOVARIETIESANDUNIRATIONALITY17Bytheprecedingdiscussion,YisdenedbyasectionofO~(d�1):(a')DenethelocusofdenitionofY;DEF(Y);tobethemaximalopensubsetof(l+1)-planes2forwhichY~hasdimensionl;i.e.isahypersurface.IfDEF(Y)isnon-empty,thenYDEF(Y)!;willbecalledtheresidualfamilyofhypersurfacesattachedto(�;X;P):(b)Thesecondaryresidual-schemeZwillrefertothescheme-theoreticintersectionZ:=Y\�:(b')WriteDEF(Z)forthemaximalopensubsetoverwhichZDEF(Z)!hasberdimensionl�1;i.e.isahypersurface.Togive

23 averyroughsummary:therstofthese,Y;i
averyroughsummary:therstofthese,Y;isthefamilyofintersectionsofXwiththefamilyof(l+1)-planescontainingthefamilyofl-planes�X,residualto�.Thesecond,DEF(Y);istheopensubsethavingnontrivialbersinY:Thethird,Z;isthefamilyofintersectionsofYwiththebasefamilyofl-planes.NoticethatDEF(Z)DEF(Y):2.4.Explicitdescriptionofthebersoftheresidualvarieties.Overanindividualclosedpointb2B;thebersoftheresidualvarietiesarereadilydescribableassubvarietiesofPb=Pr.Todothis,weworktemporarilyoverthebaseB=Spec(K):Choosehomo-geneouscoordinatesfV0;:::;Vl;Wl+1;:::;WrgonP=Prsothat�=Pl=fWi=0g:ThenfWigarehomogeneouscoordinateson=Pn�l�1:WritethedeningequationofX(=Xb)asF(V;W)=X0jIjd�1VIFI(W):HereeachIisamulti-indexofdegreejIj;andeachFI(W)isahomogeneouspolynomialofdegreed�jIj1:Now,givenapoint[wl+1;:::;wr]2;wemayparametrizethecorrespondingplaneas=f[V0;:::;Vl;wl+1U;:::;wrU]g:Wethushavehomogeneouscoordinates[V0;:::;Vl;U]onsuchthat�istheplanedenedbyU=0:TherestrictionofFtothe(l+1)-planeisgivenbyF(V;W)j=XVIUd�jIjFI(wl+1;:::;wr):Since0

24 0;jIjd;wemaydividethroughbyUtoobtainthed
0;jIjd;wemaydividethroughbyUtoobtainthedeningequationfortheprimaryresidual-schemeY;FY=F(V;W)j=U=XFI(wl+1;:::;wr)Ud�jIj�1ZI:Thescheme-theoreticintersectionofYwith�isgivensimplybysettingU=0;bywhichweobtainthedeningequationforthesecondaryresidualschemeZ;FZ(V;)=XjI0j=d�1FI0(wl+1;:::;wr)ZI0:Here,theFI0arehomogeneouslinearforms.Fromtheseequations,weseethatthelocusPn�l�1�DEF(Y);i.e.thesetofwhichareequaltoY;isthecommonzerolocus FANOVARIETIESANDUNIRATIONALITY192.5.Numbers.Finally,weintroducetherelevantnumericalboundsofwhichwewillmakeuseinthefollowingsection.DeneM(d;l)=l+d�1d�1+l�1:Thisnumberhasthefollowinguse:Lemma2.11.Letd;lk0;andnM(d;l)benon-negativeintegers,andletXPnbeak-planecontainedinahypersurfaceXofdegreed:Thenthereexistsanl-plane�suchthat�X:Proof.SinceM(d;l0)M(d;l)foralll0l;itissucienttoassumek=l�1:ChoosehomogeneouscoordinatesV0;:::;Vk;Wk+1;:::;WnonPnsuchthatthek-plane=fWi=0g:Inthesecoordinates,thedeningpolynomialofXisoftheformF(V;W)=X0jIjVIFI(Wk+1;:::;Wn):Thissumrange

25 soverthe�k+1+d�1d�1
=�k+d
soverthe�k+1+d�1d�1
=�k+dk+1
monomialsVIofdegreelessthand�1;andeachFIarehomogeneousofdegreed�jIj1intheWi:Now,giventhatn�kM�k�k+dk+1
;thepolynomialsFIhaveanontrivialcommonroot[Wk+1;:::;Wn]:Thusthespan,�;ofandthepoint[0;:::;0;Wk+1;:::;Wn]isanl-planesuchthat�X;asdesired.Nextwexanintegerk0anddenetwofunctionsrecursively,N0(d;k)andN(d;k):BeginbysettingN(2;k)=N0(2;k)=k+12+3:Thenford3;denerecursivelyN0(d;k)=M(d;N(d�1;k)+1)andthenN(d;k)=N0(d;k)+k+dd+2;orinotherwords,N0(d;k)=M(d;N0(d�1;k)+k+d�1d�1+3):Notethatbothofthesefunctionsarestrictlyincreasingwithk:Proposition2.12.(a)Inthecased=2;notethatN0(2;k)M(2;k):SofornN0(2;k):SoaquadrichypersurfaceQPnissweptoutbyk-planes.ThisinturnimpliesthatifQissmooth,thendim(Fk(Q))=(n;2;k):(b)Foralld3andk0;N0(d;k)k+dd+3k+1M(d;k); FANOVARIETIESANDUNIRATIONALITY21intersectionsinthesmoothvarietyG(k;n)([13]17.24),ifacomponentFofFk(X)meetsG(k;H)then(6)codim(FG(k;n))codim(Fk(X)G(k;n))codim(Fk(

26 Y)G(k;H))=k+dd:Itremainst
Y)G(k;H))=k+dd:ItremainstoshowthateverycomponentofFk(X)meetsG(k;H):ThiswillfollowfromastandardresultintheintersectiontheoryofGrassmannians,whichwestatewithoutproof:Lemma2.15.IfX1andX2aresubvarietiesofG(k;n)ofcodimensionc1andc2suchthatc1+c2n+1�2k;thentheintersectionX1\X2isnonempty.Notethatinthecasek=0;thisgivesthestandardintersectioncriteriononPn:Now,recallthatFk(X)iscutoutby�k+dd
conditions,henceanycomponentF0hascodimensionatmostthatnumber.SothecodimensionsofF0andG(k;H)G(k;n)areatmost�k+dd
andk+1;respectively.Wehaven+1�2kN0+1�2kk+dd+3k+2�2k=k+dd+k+2byProposition2.12(b),andsobythelemmaFmeetsG(k;H);whichestablishes6andtheclaim.(b)Assumetheresultofpart(a)forthegivenpaird;k;andletXbeahypersurfaceofdegreedinPn;withnN(d;k):Ifcodim(XsingX)N0thenpart(a)applies;otherwisedim(Xsing)n�N0N(d;k)�N0=�k+dd
+2;bydenitionofN(d;k):So,dim(Fk(X))G(k;n)=(n;d;k)+k+dd(n;d;k)+dim(Xsing)�1:Remark2.16.Foranygivenn;d;k;thesecorollariescanbededucedfromthetheoremasappliedonlytothesen;d;k:Thuswewillbe

27 abletousethecorollariesinconjunctionwith
abletousethecorollariesinconjunctionwithourinductionhypothesis.2.6.1.ABertiniLemma.Wewillneedthefollowinglemmatocontrolthesingularitiesofourhypersurfaces.Lemma2.17.(Bertini'sTheoremforaprojectivespace)LetD=fDg2PmbealinearsystemofhypersurfacesinPn;withbaselocusBPnofdimensionb:DenethesubsetsSk=fD2Djdim((D)sing)b+kg:ThenSkisaprojectivevarietyandcodim(SkPm)k: FANOVARIETIESANDUNIRATIONALITY23Writef=[F0;:::;Fm]sothattheFiarehomogeneouspolynomialswithnocommonfactorwhicharesimultaneouslyzeronowhereoutsideofB(See[15]vol.Ich.3).TheprojectivevarietydenedbytheFihasdimensionatmostb:Theberover(say)p=[1;0;:::;0]thenisdenedbyF1=


=Fm=0:ThesectionofthisberbythehypersurfacefF0=0ghasdimensiononeless(sincetheFihavenocommonfactor),andiscontainedinB:Sotheberoverp;denedbyF1=


=Fm=0;hasdimensionatmostb+1:Thisestablishes(9),whichimplies(7)andhencethemaininequalityofthelemma.2.6.2.ProofofTheorem2.13.Theorem1.6(c)ofChapter1assertsthatincase(n;d;k)�0andd3;everyhypersurfacecontainsafamilyofk-planesofdimensionatleast(theproofinvoke

28 dtheresultof[1]).Soitremainstoshowthatas
dtheresultof[1]).Soitremainstoshowthatasmoothhypersurfaceofsucientlyhighdimensionhasafamilyofk-planesofdimensionatmost:Tothisend,wewillconstructaconvenientfamilyofpairsofplanes,
:WewillthenndalowerboundL(step1)andanupperboundU(step2)ondim(
):TheresultinginequalityLUwillsimplifytothedesiredupperboundondim(Fk(X))(step3).TheboundsLandUwillbeestablishedbyinductionond:Proposition2.12(a)exactlyestablishesthebasecased=2ofthetheorem,forallk:Fortheinduction(theremainderoftheproof),xdandk;andassumethatTheorem2.13anditscorollariesholdforalltriples(n0;d0;k0)withn0n;d0d;k0kbutnotallequal.Notethatthetheoremisvacuousforn0N0(d0;k0).LetXPn;withnN0(d;k);beasmoothhypersurfaceofdegreed:Setl=N(d�1;k);anddenethelocallyclosedsubvariety
Fk(X)Fl(X)givenby
=f(;�)2Fk(X)Fl(X)jdim(\�)=k�1g:Equivalently,
canbedenedbytheconditionthatand�togetherspanan(l+1)-planeinPn(notnecessarilycontainedinX).Remark2.18.Theideaisroughlyasfollows:ournumbersaresucientlyhighthatnotonlywillXbesweptoutbyk-planes,buteachofthesek-planesinXiscontainedinanl=

29 N(d�1;k)-planeinX:Thiswillallowustoap
N(d�1;k)-planeinX:Thiswillallowustoapplyourinductionhypothesisondrsttoshowthat
isdenseinaberproductof ag-Fanovarieties(step1),thentoinject
intotherelativeFanovarietyofthemainresidualschemeYcomingfromtheintersectionofXwiththevarietyof(N(d�1;k)+1)-planescontainingitsFanovarietyofl=N(d�1;k)-planes(step2).Step1.LetFk�1;k(X)andFk�1;l(X)bethe ag-FanovarietiesFk�1;k(X)=f( ;)j XgG(k�1;n)G(k;n)andFk�1;k(X)=f( ;�)j �XgG(k�1;n)G(l;n):Wewillrealize
asanopen,densesubsetoftheberproduct:=Fk�1;k(X)Fk�1(X)Fk�1;l(X)Fromthedenition,thereisaregularmap
!Fk�1(X) FANOVARIETIESANDUNIRATIONALITY25where�= ;�isthe(l+1)-planetheyspan.NotethatY�asrequired:theopendenseset��iscertainlycontainedintheresidualvarietyY�;whichisclosedandhencecontains:Thismapisclearlyinjective.Soinwhatfollows,wewillsimplycomputeanupperboundonthedimensionofFk(Y=):ThiswillfurnishourupperboundUonthedimensionof
:Todothis,xanl-plane�0X;andwewillcomputethedimensionofthe

30 2;berofFk(Y=)over�0:Forthispurpos
2;berofFk(Y=)over�0:Forthispurpose,weneedonlyconsidertheresidualfamilyoverthesinglel-plane�0;i.e.Y0:=fYj2�0=Pn�l�1g:SowewouldliketocomputethedimensionofFk(Y=)�0=Fk(Y=�0)=Fk(Y0=Pn�l�1):TheYhavedegreed�1;sowehopetoapplytheinductionhypothesisonceweknowsomethingabouttheirsingularities.Happily,sinceXissmooth,byProposition2.9thehypersurfacesfZ=Y\�0gformabase-point-freelinearseriesin�0|andsobyProposition2.17,thelocusS=f2DEF(Z)�0Pn�l�1jdim((Z)sing�1ghascodimensionatleastinPn�l�1:Hence,thevarietyW=f2DEF(Z)�0DEF(Y)�0jdim((Y)singgSmustalsohavecodimensionatleastinPn�l�1:Thuswecanchoose2DEF(Z)�0�W1;andthehypersurfaceYhasatmostisolatedsingularities.So,sincel=N(d�1;k);theinductionhypothesisyieldsfromCorollary2.14thatdim(Fk(Y))=(l+1;d�1;k):WemaythusconcludethattheinverseimageofDEF(Z)intheberFk(Y0=Pn�l�1)=Fk(Y=)�0hasdimension(l+1;d�1;k)+n�l�1:Itremainstocheckthattheinverseimageofthep-dimensionalprojecti

31 velinearspace(DEF(Z)�0)c=Pn�l�1
velinearspace(DEF(Z)�0)c=Pn�l�1�DEF(Z)�0(asperProposition2.9)doesnotintroducecompo-nentsoflargerdimensioninFk(Y=)�0:SincetheseriesfZgisbase-pointfree,wemusthavel+1codim((DEF(Z)�0)cPn�l�1);sincethisisthedimensionofthelinearsys-temfZg:Meanwhilewehavethetrivialupperbounddim(Fk(Y))dim(G(k;l+1));sotheberdimensioncanjumpfromthegenericdimensionbyatmostdim(G(k;l+1))�(l+1;d�1;k)=�k+d�1d�1
:Butsincethelocus(DEF(Z)�0)chavingpossiblyhigher-dimensionalbershascodimensionatleastl+1�k+d�1d�1
;theseberscannotcontributeahigher-dimensionalcomponentofFk(Y=)�0:Wecanconcludethatdim(Fk(Y=)�0)=(l+1;d�1;k)+n�l�1:Since�02Fl(X)wasarbitrary,wemayconcludenallythatdim(
)dim(Fk(Y=))=dim(Fl(X))+(l+1;d�1;k)+n�l�1=dim(Fl(X))+(k+1)(l�k�1)��k+d�1d�1
+n�l�1=:U 3.UnirationalityofSmoothHypersurfacesofLowDegree.InthischapterwewilldemonstrateanimportantconsequenceofTheorem2.13:theunirationalityofsmooth,low-degreehypersurfaces.Ourmainreferencewillagainbe[2].InS

32 ection3.3,however,wearriveatsimilarconst
ection3.3,however,wearriveatsimilarconstructionstothoseof[2]muchmorestraightforwardly.Themethodsofthischapterwereoriginallydevelopedforthecaseofageneralhypersurfaces.TheconclusionofthelastchapterwillbeusedtoproveCorollary3.12:weareabletoboundthedimensionofthevarietyofplanescontainedinthesmoothbersinorderto\in ate"theimage,showingsurjectivity.Note:Allbersofmorphismswillnowbe\scheme-theoretic,"aswillallinclusions.A\point,"however,willstillreferalwaystoaclosedpoint.Example3.1.OurmethodofproofinthischaptergeneralizesthatofProposition2.3(areviewofthisexampleisrecommended).Webeginwithaheuristicattempttoprovethecaseofquartichypersurfaces,identifyingthemainobstructionstoitssuccess.LetXPnbeasmoothquartichypersurface.HopingtoextendtheproofofPropo-sition2.3,wetakensucientlylargethatwecanchooseanl-plane�Xandconsiderthe(l+1)-planesthatcontainit.AgeneralsuchplaneintersectsXintheunionofaresidualcubichypersurfaceYandtheplane�:Asbefore,theblow-up�:~X=Bl�(X)�!Pn�l�1=hasgeneralberacubichypersurfaceYPl+1:Now,weexpectthatageneralberYof~X!wi

33 llbesmooth.7Inthiscase,Ywillbeunirat
llbesmooth.7Inthiscase,Ywillbeunirational,byProposition2.3.However,ifwehopetousethisfact,wemustbeabletochooseak-plane oneachcubicY:Asbefore,wecannarrowdownthechoicebylookingonlywithinthel-plane�;whichmeetsYinthesecondaryresidualschemeZ=�\Y:Asinthecaseofcubics,wedonotexpecttochooseauniquesuchplaneforeachZ;i.e.arationalsection.Insteadwesetuptheincidencecorrespondence :=f(; )j Z=Y\�gPn�l�1G(k;�):(Forcubics,wehadk=0sothatG(0;�)=�:)Asbefore,weformthepullbackof~X!=Pn�l�1tothisvariety !;H:=~X :ThefamilyH1! isthenak-planedfamilyofcubichypersurfaces.ThisgivesusaunirationalparametrizationofeachcubicY:AndweshouldbeabletofurtherpullbackH1toobtaina\pointed,k-planedfamilyofhypersurfaces"H0;dominatedbyavarietywhichwillberational|providedthat itselfisrational.(SeeSection3.4.1forabetterdescription.)Thequestionofwhether isrationalcanbeapproachedasbefore,byconsideringtheprojectionmaptothesecondfactorG(k;�):AsinProposition2.9,thesecondaryresidualhypersurfacesfZgformalinearsystem,andeachpoint �imposesacertainnumberoflinearco

34 nditionson2:Thuswemayhopethat is
nditionson2:Thuswemayhopethat isaprojectivebundleovertherationalbaseG(k;�);andwillhenceberational. 7Infactitwillbesucientthatthegeneralberhaveatworstisolatedsingularietieswhennd;asintheproofofTheorem2.13.27 FANOVARIETIESANDUNIRATIONALITY29(ThefactT=n=Tn=nnis[19]Prop.5.1.5.)Theinverseimage\x"ofxintheberXycorrespondstotheideal(m 1)(S TK)=m TK:Furthermore,thelocalring(S TK)m TKisnaturallyisomorphictoSm TK;underwhichthemaximalidealcorrespondstomm TK|thisisreadilyseenfromexaminingthesurjectiveringhomomorphismS!S T1;whichexpressesS TKS=((n)m);andfromtheexactnessoflocalization([16]Ch.3).Weprovetheexactnessofthedualsequencetotheonestated,TyY�!TxX�!TxXb�!0(10)nn=n2nx�!mm=m2m�!(mm=m2m) TK�!0:(11)Wemaywriteout(mm=m2m) TK=(mm TT)=(m2m TT+mm Tn)mm=(m2m+(n)mm):But(n)mm=x(nn)mminthelocalringOX;x=Smsinceanyelements62nissentto(s)62m:Thustherightsideofthelastequationisequalsimplyto(mm=m2m)=(x(nn));whichisjustthedesiredstatementthat(10)isexact.Proposition3.3.LetXBPnbeafamily,withXandBvarieties.Ifthescheme-theoreticber

35 sXboverclosedpointsb2Barek-dimensionalli
sXboverclosedpointsb2Barek-dimensionallinearsubspacesofPn;thenXisaprojectivebundleofrankkoverB;i.e.islocallytrivial.Note.TherequirementthatXbeasub-bundleofatrivialbundle,i.e.XBPn;isessential,asperRemark2.2.Proof.Givenb02BaclosedpointandXb0=Pkaber,choosean(n�k�1)-plane=Pn�k�1PnsuchthatXb0\=;:LetUbetheopensubsetfb2BjXb\=;g.Let:Pn�!Pkbetheprojectionfrom:ThenIdisbijectivebetweentheclosedpointsofXU=�11(U)andUPk(linearalgebra).From[13]Corollary14.10,itremainstoshowthatIdhasinjectivedierentialatallclosedpoints(b;p)2XU:Formthediagram0// T(b;p)Xb//  T(b;p)XU// (Id)(b;p) TbU//  00// T(b;(p))(fbgPk)// T(b;(p))(UPk)// TbU// // 0;where(ifthisisnotobvious)theleftverticalmapisderivedfromthesequencePk=Xb!XU!UPk!Pk!fbgPk;whichisanisomorphismsincenonconstant.Theleftandrightverticalmapsinthediagramarethusisomorphisms,thereforethecentralmapisanisomorphism.Alternatively,noticethat[14],III-56directlyimpliesthatX!Bisa atfamily.SoXisaberproductw

36 iththeGrassmannianbundle,henceabundleits
iththeGrassmannianbundle,henceabundleitself. FANOVARIETIESANDUNIRATIONALITY31Corollary3.10.IfasmoothhypersurfaceXPnkofdegreedcontainsanL(d)-planethatisrationaloverk;thenXisunirationaloverk:Asanexampleofhowthismightwork,considertheresultsofSection3.1above.NotethattheL(d)-planeXisrationaloverkifandonlyiftheresidueeldateachclosedpointx2L(d)isequaltok|andinthiscase,thetangentspacesatxand(x)arenite-dimensionalk-vector-spaces,andthestatement(andproof)ofLemma3.2makessense.3.3.Constructions.Inthissectionwemakefurtherconstructionscorrespondingtoagivenl-planedfamilyofhypersurfaces(�;X;P)overB:ThesewillformalizethetechniquesofProposition2.3andExample3.1.Weassumethattheschemes;Y;Z;etc.,ofSection2.3.4correspondtoourgivenl-planedfamily.LetG:=Grassk(�)bethefamilyofk-planescontainedintheprojectivebundle�:ThereisamapGrassk(�)!GcomingfromthemapofB-schemes(12)Grassk(�)�!Grassk(�)B�!Grassk(�)=G;fromwhichwegetanothermap :Fk(Z=),!Grassk(�)�!G:Now,eachberofFk(Z=)!Goverak-planeb2Gbisinfactalinearsubspaceofb=Pn�l�1:Thisfo

37 llowsformProposition2.9ofSection2.4:each
llowsformProposition2.9ofSection2.4:eachpointof�bsimplyimposesalinearconditiononthelinearseriesfZbgb2b;sothepointsofbmerelyimposesomanylinearconditionsontheparametersb2b.(ByapplyingthedenitionoftherelativeFanovarietytothediscussionofSection2.4,oneisreadilyconvincedthattheselinearconditionscutouttheberinFk(Z=)overbscheme-theoretically,sothattheberisareducedlinearspace.)However,wecannotbeassuredthatallofthebershavethesamedimension;althoughgenerically,theydo,bytheTheoremonFiberDimension(1.3).LetCbetheopensubsetoftheirreducibleprojectivevarietyGoverwhichthebersofthismap havetheminimumpossibledimensionl��k+d�1k
;i.e.thesetofk-planesimposingindependentconditionsonthelinearseries.Atpresentwedonotknowthatthisisnonempty;thiswillfollowfromCorollary3.12,ifthedimensionlisappropriatelyhigh.Thus,fromLemma3.3,therestrictiontoCofFk(Z=)!GisinfactaprojectivebundleoverC,whichwewilldenotebyQ:Fk(Z=)C=:Q�!CG:WealsohaveamorphismQ�!;sinceQisasub-bundleoftherestrictiontoCofthePn�l�1-bundleGB�!G:ApointofthevarietyQoveraclosedpointb2Bcan

38 bethoughtofasapair( b;b)2Grassk(�
bethoughtofasapair( b;b)2Grassk(�b)b;suchthatsuchthat bZb|inotherwords, bZb�bbPb;whichalsoimpliesthat bYbb FANOVARIETIESANDUNIRATIONALITY333.4.Proofofunirationalityofsmoothlow-degreehypersurfaces.InSection3.4.2,wewillshowthatforafamilyofhypersurfacesX!Bofappropriatelyhighdimensionrelativetoitsdegreed(seesection3.2),thenaturalmapYD!Xisdominant.ThedegreeofthefamilyofhypersurfacesYDisequaltod�1;whichwewillusetoinductondegreeinSection3.4.3.3.4.1.Roughdescriptionoftheproof.Intheinductionargumenttofollow,thelocationoftheparametrizingvarietyitselfisobscured.ItisthereforeinstructivetogivethefollowingroughdescriptionoftheproofofTheorem3.6:ApplyProposition3.11toobtainsuccessivefamiliesofhypersurfacesYd�iDd�iofdegreed�1;d�2;d�3:::;witheachfamilydominatingthepreviousones;theirbasesDd�iwillformacombovertheoriginalbaseB;providedtheDicanbechosennonempty(wewillestablishthisusingthedimension,degree,andsmoothnessrequirementsofthetheorem).Thiscanbedrawnoutasfollows:X Yd�1Dd�1 oo Yd�2Dd�2 oo :::oo Y1D1=D0&#

39 15; oo BDd�1oo Dd�2oo :::oo D
15; oo BDd�1oo Dd�2oo :::oo D1:oo WewillarriveatafamilyofhypersurfacesY1D1ofdegreed=1dominatingallpriorfamiliesYd�iDd�i.ByProposition3.3,thislastfamilyY1D1isthenitselfaprojectivebundleoverD1;whichwemaycallD0:HenceD0!D1!:::!BwillbeacomboverB(ifnon-empty),withadominantmapD0!XgivenbythecompositionofdominantmapsD0!Y2D2!


!X:ToobtainCorollary3.7,simplyobservethatiftheoriginalbaseBisrational,thenthebasesDiateachsteparerational(seeProposition3.5).ThisincludesthelastfamilyD0;whichisrationalandwilldominateallthepreviousvarietiesYiDiandX:HenceXwillbeunirational.IncaseB=Spec(K)andXisasinglehypersurface,theparametrizingvarietyisacertainsubvarietyofacross-productof agmanifoldsinPr(see[2]foranexplicitdescription).3.4.2.DominanceofYD!X.WeproceedtoestablishtheremainingfactswhicharenecessarytoshowthatYD!Xisdominant.TheseareobtaineddirectlyfromTheorem2.13ofthepreviouschapter.Corollary3.12.Letdandkbepositiveintegers,andletN=N(d;k)asdenedinSection2.5.LetlN(d;k)beaninteger,andletD=fDPlg2Pmbeabase-point-freelinearseriesofhypersurfacesofdegreedinPlparametrizedbyPm(necessarily,ml).Let =f(

40 ;)2PmG(k;l)jD
;)2PmG(k;l)jDgbetheincidencecorrespondence.Then, FANOVARIETIESANDUNIRATIONALITY35bereducible.Buteachset-theoreticberoverG(k;l)isaprojectivespace,soauniquecomponentof dominates.Now,assumegivenanl-planedfamilyofhypersurfaces(�;X;P)overanintegralbaseB;andassumethattheconstructionsofsection3.3havebeenmadeforthisfamily.Proposition3.13.IfX!Bissmoothalong�andlN(d�1;k);thenDisnonempty,henceacomboverB;andthenaturalB-morphismYD!Xisber-by-berdominantoverB(andthereforedominant).Proof.(Itmaybehelpfulheretorefertothediagram(14).)Asbefore,itwillsucetoworkoverasingleclosedpointb2B;soweassumetemporarilythatB=Spec(K):ThusX=Xbissmoothalong�=�b=Pl;andbyProposition2.9(b),thesecondaryresidualschemesfZgformabase-pointfreelinearseriesofdegreed�1in�(abasepointwouldbesingular).ThereisatrivialtechnicalitycomingfromProposition2.9:thelinearseriesfZgisparametrizednotby=Pr�l�1butbyaquotientof;whichwewillcallPm:ThereisthusasurjectivemapDEF(Z)PmwhosebersareprojectivesubspacesPpDEF(Z)ofrankp=r�l�m�1:Thismapinducesasurje

41 ctiverationalmapfromtheFanovariety(
ctiverationalmapfromtheFanovariety(overb),Fk(Z=)(Grassk(�)B)b=G(k;l)Pn�l�1;ontothetheincidencecorrespondence\ "G(k;l)PmofCorollary3.12:Fk(Z=)////____ $$IIIIIIIIII zzuuuuuuuuuuu @@@@@@@@@G=G(k;l)////___Pm:EachberofthismapisofcoursealsoaprojectivespacePp:Now,byCorollary3.12(a),sincelN(d�1;k)�k+d�1k
;thegeneralk-planeimposesindependentconditionsonthelinearseriesfZg;andeachberisnon-empty.Thus !Gissurjective,asisFk(Z=)!G;andtheopensubsetCGisnon-empty.So,theprojectivebundleQFk(Z=)isanon-emptyopensubset.Ontheotherhand,theopensubset0isalsonon-empty(overb):for,DEF(Z)bisnon-emptybyCorollary3.12(b),andthegeneralZissmoothbyLemma2.17.SoDQ;theinverseimageof0:Therefore,followingProposition3.11,D!C!Bisinfactacomb.WenowshowthatYD!Xdominatesoverb.ByCorollary3.12(b),\ "dominatesPm;henceFk(Z=)dominates=Pr�l�1asdotheopensubsetsQandD:Let10beanopensubsetcontainedintheimageofD:LetX0Xbetheopensubsetlyingoutsideofboth�andtheinverseimageof�1underthepro

42 jection�:X��!;thissetX
jection�:X��!;thissetX0isnon-emptysince10DEF(Z)DEF(Y):Furthermore,X0isdense,sinceX=Xbmustbeintegralinordernottomeet�inasingularpoint.Letx= x;�:Chooseak-plane xZx;so( x;x)isaclosedpointofD(possiblesincelN(d�1;k)).Thenbecausex62�;wehavex2Y( x;x):ThusxisintheimageofY( x;x);andwearedonesincex2X0wasgeneral. FANOVARIETIESANDUNIRATIONALITY37References[1]M.HochsterandD.Laksov,LinearSyzygiesofGenericForms,Comm.inAlgebra15(1,2),227-239(1987).[2]J.Harris,B.Mazur,andR.Pandharipande,HypersurfacesofLowDegree,DukeMath.J.95,No.1(1998),125-160.[3]O.Debarre,LinesonSmoothHypersurfaces,preprint,http://www-irma.u-strasbg.fr/debarre/.[4]J.M.Landsberg,DierentialGeometryofSubmanifoldsofProjectiveSpace,preprint,http://arxiv.org.[5]R.Beheshti,LinesonProjectiveHypersurfacesJ.reineangew.Math,2006.[6]K.ParanjapeandV.Srinivas,UnirationalityoftheGeneralCompleteIntersectionofSmallMultidegree,Asterisque211,Soc.Math.France,Montrouge,1992.[7]U.Morin,\Sull'unirationalitadell'ipersuperciealgebricadiqualunqueordineedimensionesuciente-mentealta,"AttiSecondoCo

43 ngressoUn.Mat.Ital.,Bologna,1940,Edizion
ngressoUn.Mat.Ital.,Bologna,1940,EdizioniCremorense,Rome,1942,298-302.[8]Zariski,Oscar,OnCastelnuovo'scriterionofrationalitypa=P2=0ofanalgebraicsurface,IllinoisJ.Math.2(1958),303{315.[9]J.Starr,FanoVarietiesandLinearSectionsofHypersurfaces,preprint,http://arxiv.org(2006).[10]J.Harris,M.Roth,andJ.Starr,RationalCurvesonHypersurfacesofLowDegree,preprint,http://arxiv.org(2002).[11]X.Chen,UnirationalityofFanoVarieties,DukeMath.J.90No.1,63-71(1997).[12]J.Kollar.LowDegreePolynomialEquations:Arithmetic,GeometryandTopology,preprint,http://arxiv.org(1996).[13]J.Harris,AlgebraicGeometry:aFirstCourse,GTM133,NewYork,Springer-Verlag,1992.[14]D.EisenbudandJ.Harris,TheGeometryofSchemes,GTM197,NewYork,Springer-Verlag,2000.[15]I.R.Shafarevich,BasicAlgebraicGeometryvols.1and2,NewYork,Springer-Verlag,1994.[16]M.AtiyahandI.G.Macdonald,IntroductiontoCommutativeAlgebra,London,Addison-Wesley,1969.[17]R.Hartshorne,AlgebraicGeometry,GTM52,NewYork,Springer-Verlag,1977.[18]D.Eisenbud,CommutativeAlgebrawithaViewtowardAlgebraicGeometry,GTM133,NewYork,Springer-Verlag,1994,http://books.google.com.[19]P.Samuel,AlgebraicTheoryofNumbers,Boston,HoughtonMi&