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Advances in Mathematics
www.elsevier.com/locate/aim
Generalized Lazarsfeld–Mukai bundles and a conjecture of Donagi and Morrison
✩Margherita Lelli-Chiesa
MaxPlanckInstituteforMathematics,53111 Bonn,Germany
a r t i c l e i n f o a b s t r a c t
Articlehistory:
Received9October2013 Accepted4August2014 Availableonline17October2014 CommunicatedbyRaviVakil
Keywords:
K3 surfaces Brill–Noethertheory
Let S be a K3 surface and assume for simplicity that it does not contain any (−2)-curve. Using coherent systems, we expressevery non-simple Lazarsfeld–Mukaibundle on S asanextensionoftwosheavesofsomespecialtype,thatwe refer to as generalized Lazarsfeld–Mukai bundles. This has interestingconsequencesconcerningtheBrill–Noethertheory ofcurvesClyingonS.Fromnow on,letgdenotethegenus of C and A be a complete linear series of type gdr on C such that d ≤ g−1 and the corresponding Brill–Noether number is negative. First, we focus on the cases where A computes the Clifford index; if r > 1 and with only some completely classified exceptions, we show that A coincides with the restriction to C of a line bundle on S. This isa refinementofGreenandLazarsfeld’sresultontheconstancy of the Clifford index of curves moving in the same linear system.Then,we studyaconjectureofDonagiandMorrison predicting that, under no hypothesis on itsClifford index, Aiscontained ina ges which iscutout from a linebundle onSandsatisfiese≤g−1.We providecounterexamplesto thelast inequalityalreadyfor r = 2. A slight modification oftheconjecture,which holdsfor r = 1,2,is proved under some hypotheses on the pair (C,A) and its deformations.
We show that the result is optimal (in the sense that our hypothesescannotbeavoided)by exhibiting,inAppendix A,
✩ Withanappendix jointwithAndreasLeopoldKnutsen.
E-mailaddress:[email protected].
http://dx.doi.org/10.1016/j.aim.2014.08.011 0001-8708/© 2014ElsevierInc.All rights reserved.
somecounterexamplesobtainedjointlywithAndreasLeopold Knutsen.
© 2014ElsevierInc.All rights reserved.
1. Introduction
The use of Lazarsfeld–Mukai bundles (LM bundles for short) in the study of the Brill–Noether theory ofcurvesC lyingonaK3 surfaceS hasbrought severalachieve- ments,suchasanewproofoftheGieseker–PetriTheorem[13],theclassificationofprime Fano manifolds of coindex 3 [16], Green’s Conjecture for a general curve of any given genusg[19,20].Morerecently,therehavebeenapplicationstohigherrankBrill–Noether theory (cf. [6])and hyperkählermanifolds(cf. [3]).Thekey observationis that,ifC is general in its linear system, non-trivial endomorphisms of the rank r+ 1 LM bundle EC,(A,V)associatedwithalinearseries(A,V)∈Grd(C) measurethefailureofinjectivity of thePetri map
μ0,(A,V):V ⊗H0
C, ωC⊗A∨
→H0(C, ωC);
if ρ(g,r,d) <0 with g equalto the genus of C, then the hypothesis on the genericity of C is not necessary. For later use, we recall that inthe case of acomplete grd onC, that is,whenV =H0(C,A),thecorresponding LMbundleisdenotedbyEC,A.
The study of non-simple LM bundles turns out to be crucial. If r = 1, these arise as extensions of two torsion free sheaves of rank 1 on S which are the image and the kernel of an endomorphismdropping the rankeverywhere[4]. Non-simple LMbundles ofrank 3 (i.e.,satisfyingr= 2)wereinvestigatedin[14]byusingthefactthattheycan- notbestable andlookingatthecorrespondingHarder–NarasimhanandJordan–Hölder filtrations.
Ourideainordertotreatnon-simpleLMbundlesEC,Aofarbitraryrankistoconsider the pair (EC,A,H0(S,EC,A)) as a coherentsystem àla Le Potier [15]; if EC,A is non- simple,thenthesameholdsforthepair(EC,A,H0(S,EC,A)).Thenotionofstabilityfor coherentsystems dependsonthe choiceofapolynomialα∈Q[t] withpositiveleading coefficient, andnon-simplicityimpliesthenon-stabilitywithrespectto everyα.Having fixedαequaltotheHilbertpolynomialpS ofOS,we considerthemaximaldestabilizing sequence of(EC,A,H0(S,EC,A)).
A firstapplicationof ourmethodsconcernscomplete linearseries Aof typegdr ona smooth curve C ⊂S, whenever Acomputes theCliffordindex ofC and ρ(g,r,d)<0.
SetL:=OS(C);in orderto stateourresult,we recallthatalinebundleM∈Pic(S) is adapted tothelinear system|L|exactlywhen:
(1) h0(S,M)≥2 andh0(S,L⊗M∨)≥2;
(2) Cliff(M⊗ OC) isindependent ofthecurve C∈ |L|.
Condition (1) makes sure thatM⊗ OC contributes to the Clifford index,while (2) is satisfiedifeitherh1(S,M)= 0, orh1(S,L⊗M∨)= 0.
Weprovethefollowing:
Theorem 1.1. Let A be a complete gdr on a non-hyperelliptic and non-trigonal curve C⊂S suchthat d≤g−1,ρ(g,r,d)<0and Cliff(A)= Cliff(C).AssumeL:=OS(C) isample andthefollowingconditionissatisfied:
(∗) there is no irreducible ellipticcurve Σ ⊂S such that Σ·C = 4 andno irreducible genus 2curveB ⊂S suchthat B·C= 6.
Then, oneof thefollowingoccurs:
(i) ThereexistsalinebundleM ∈Pic(S)adapted to|L| suchthatA M⊗ OC. (ii) The line bundle A satisfies h0(C,A)= 2 (i.e., r= 1); furthermore, there exists a
linebundle M ∈Pic(S) adapted to|L| such that |A| iscontained inthe restriction of |M|toC.
Inparticular, as soon as d≤g−1 and r >1, theline bundle A coincides with the restriction to C of a line bundle M ∈ Pic(S). In case (ii), the condition that |A| is containedintherestrictionof|M|toCmeansthatforeveryA0∈ |A|there isadivisor M0∈ |M|suchthatA0⊂M0∩C;ifH0(S,M) H0(C,M⊗ OC),thisisequivalentto therequirementh0(C,A∨⊗M⊗ OC)>0.
Ifcondition(∗)isviolated,theredoexistexceptionswhereneither (i)nor (ii)happen;
we actuallyproveastrongerversionofTheorem 1.1(cf.Theorem 4.2),whichalsocovers andcompletelyclassifiessuchexceptionalcases.Notethatourresultmakesnoassump- tionontheCliffordindexofthecurveC.As soonasL:=OS(C) isample,Theorem 1.1 canbe seenas a refinementof Green and Lazarsfeld’sresult thatallsmooth curves in
|L|havethesameCliffordindex(cf. [10]).
Theexceptionalcasesmentionedaboveprovidecounterexamplestothefollowingcon- jectureof DonagiandMorrison:
Conjecture1.2. (See[4,Conjecture 1.2].)LetC beasmoothcurve ofgenus g≥2lying onaK3surfaceS andletA beacomplete,base pointfreegdr onC suchthat d≤g−1 andρ(g,r,d)<0.Then,thereexistsalinebundleM ∈Pic(S)suchthat|A|iscontained intherestriction of|M|toC andthefollowinginequalities aresatisfied:
c1(M)·C≤g−1, Cliff(M⊗ OC)≤Cliff(A).
Wedescribeonecounterexampleexplicitly(cf.Counterexample 1).Letπ:S→P2be a2 : 1coverbranchedalong asmoothsexticand assumethatB :=π∗OP2(1) generates the Picard group of S. A curve C ∈ π∗|OP2(3)| ⊂ |3B| is a 2 : 1 cover of an elliptic
curve Γ; letA be the pullback to C of OΓ(P1+P2+P3), where P1, P2, P3 are three non-collinearpoints.OnemayshowthatAisacompleteg26 onC andthelinearsystem
|A| iscontained in|OC(2B)| butnotin|OC(B)|; since2B·C > g−1,Conjecture 1.2 fails.
Notice that the above counterexample does not contradict the existence of a line bundle M ∈ Pic(S) such that the linear system |A| is contained in the restriction of
|M|toC,butonlytheinequalityc1(M)·C≤g−1.In fact,onemightstillbelievethe following modificationoftheconjecture:
Conjecture1.3.LetCbeasmoothirreduciblecurveofgenusg≥2lyingonaK3surface S and A be acomplete, base pointfree gdr on C suchthat d≤g−1 andρ(g,r,d)<0.
Then, there existsalinebundleM∈Pic(S),adapted to|L|,such that:
(i) |A| iscontainedintherestriction of|M|toC;
(ii) Cliff(M⊗ OC)≤Cliff(A).
The condition c1(M)·C ≤ g−1 is here replaced with the requirement that M is adapted to |L|. Theorem (5.1) in [4] proves both Conjecture 1.2 and Conjecture 1.3 for r = 1. The refined Conjecture 1.3 for r = 2 was obtained in [14, Theorem 1.1]
under somemild hypotheses on theline bundle L; note that, for r= 2,the inequality c1(M)·C≤(4g−4)/3 wasalsoobtainedandCounterexample 1showsthatthisbound is optimal.
ByusingcoherentsystemsandageneralizationofthenotionofLMbundles,we reduce Conjecture 1.3 toaquestionof secantvarieties.Ourmainresultisthefollowing:
Theorem1.4.LetSbeasmoothprojectiveK3surfacecontainingno(−2)-curves.LetA be acomplete, basepointfree gdr onasmooth genus g curveC ⊂S suchthat d≤g−1 andρ(g,r,d)<0,andassumethatthepair(C,A)hasnounexpectedsecantvarietiesup to deformation.Then, Conjecture 1.3 holdsforA.
Weexplainthestatementofthetheorembyspendingafewwordsonthenewconcept of havingno unexpectedsecantvarietiesup todeformation.
Given A∈Wdr(C) andhaving fixedintegers0≤f < e,thevarietyofsecantdivisors Vee−f(A) isthedeterminantal variety:
Vee−f(A) :=
D∈Ceh0
C, A(−D)
≥r+ 1−e+f .
Bydefinition,thevarietyVee−f(A) parametrizeseffectivedivisorsofdegreeeonCwhich impose at moste−f conditionsonthelinearsystem|A|. IfA isveryample, theseare the(e−f−1)-planeswhicharee-secanttotheembeddedcurveC|→A|Pr.Alsonotethat, when A = ωC, the variety Vee−f(A) is the inverse image of the Brill–Noether variety Wef(C) under theAbel–Jacobi mapCe→Pic(C).It isclassicallyknownthat:
expdimVee−f(A) =e−f(r+ 1−e+f);
we referto [5]for resultsconcerning theexistence (resp.non-existence) of linearseries withspecialsecancyconditionsonanarbitrary(resp.general)genus gcurve.
ReplacenowCwithanintegral(andpossiblysingular)curveX,andAwithatorsion free sheaf B ∈ Wrd(X), i.e., lying in the compactified Jacobian Jd(X) and satisfying h0(X,B)≥r+ 1.Then, the definitionofthe secantvarietyVee−f(B) stillmakessense withsomeslightmodifications:
Vee−f(B) :=
q∈QuoteBh0(X,kerq)≥r+ 1−e+f , wheretheQuotscheme QuoteB parametrizesquotientsq:B→Qofdegreee.
GivenCandAasinConjecture 1.3,we saythatthepair(C,A)has someunexpected secantvarieties upto deformationif itcanbe deformedto apair(X,B) suchthatthe followinghold:
• The curve X ∈ |L| is integral, the sheaf B ∈ Jd(X) is globally generated and EX,B EC,A;in particular,h0(X,B)=r+ 1.
• Forsomeintegers0≤f < e,onehasVee−f(B)=∅and expdimVee−f(B)<0.
Thebundle EX,B above isdefined inthe sameway as LM bundles for line bundles onsmoothcurves.Inotherwords,EX,B isthedualofthekerneloftheevaluationmap H0(X,B)⊗ OS B.
Inthejointappendix withAndreasLeopoldKnutsen,we showthatthehypothesisin Theorem 1.4concerningsecantvarietiescannotbeavoidedbyexhibitingacounterexam- pletoConjecture 1.3.Thefailure oftheconjectureisobtainedalongwiththeexistence ofsomeunexpectedsecantvarietiesonadeformationofthepair(C,A).
The organization of the paper is as follows. In Section 2, we introduce generalized Lazarsfeld–Mukai bundles (g.LM bundles in the sequel, cf. Definition 1), which share somecommon propertieswith LMbundles. In particular,the definitionof theClifford indexofag.LMbundleE willplayacentralroleintheproof ofTheorems 1.1 and1.4;
byCorollary 2.5, Cliff(E) is nonnegativeas soonas c1(E)2>0.Thecasec1(E)2= 0 is coveredbyProposition 2.7.
Section 3 contains some preliminaries on coherent systems. Theorem 3.4 expresses anynon-simpleand globallygenerated LMbundleas anextensionof ag.LMbundle of type (II)bytheelementarymodificationofag.LMbundleoftype (I).
In Section4, Theorem 1.1 is proved. The proof is made technically difficult by the possiblepresenceofsmoothcurvesonS whicharerationalor ellipticorhyperelliptic.
Finally, in Section 5 we focus on Conjectures 1.2 and 1.3 and prove Theorem 1.4.
Thestrategyconsistsinconsideringthemaximaldestabilizingpair(E1,H0(E1)) ofthe coherentsystem(EC,A,H0(EC,A)) andinshowingthat,as soonasthevectorbundleE1 hasrank≥2,somedeformationofthepair(C,A) hasanunexpectedsecantvariety.
1.1. Notationandpreliminaries
Forus, S will alwaysbeasmoothprojectiveK3 surfaceand Casmoothirreducible curveon it,whosegenusisdenotedbyg.A linearseries(A,V) oftypegdronCiscalled primitive ifboth (A,V) and ωC⊗A∨ arebase point free.We setL:=OS(C), andwe denotebyWdr(|L|) thevarietyparameterizing pairs(C,A) withCasmoothirreducible curve inthelinearsystem|L|andA ∈Wdr(C).
Inallcaseswherenoconfusionarises,givenasheafF onaschemeY,we willsimplify notation and write Hi(F) (or hi(F) for the corresponding dimension), dropping any reference toY.
Given a0-dimensionalsheafτ onS, we denotebyl(τ) itslength,whichcoincides by definitionwithh0(τ).
All diagrams appearing inthe paper are commutativeand allof their columns and rowsareexact.
Throughout thepaper, we will make frequentuse of the following strong versionof Bertini’sTheorem,duetoSaint-Donat.
Theorem1.5.(See[18].)LetLbealinebundleonaK3surfaceS suchthath0(S,L)>0.
Then,|L|hasnobasepointsoutsideitsfixedcomponents.Furthermore,if|L|hasnobase components, theneither of thefollowingholds:
(i) Ifc1(L)2>0,thenh1(S,L)= 0andageneralelementin|L|isasmooth,irreducible curveof genus g= 1+c1(L)2/2.
(ii) If c1(L)2 = 0, then there exista number k∈ Z>0 andan irreducible curve Σ⊂S with pa(Σ) = 1 such that L = OS(kΣ). In this case, one has h0(S,L) = k+ 1, h1(S,L)=k−1andeveryelementin|L|canbewrittenasasumΣ1+Σ2+· · ·+Σk
withΣi∈ |Σ|for1≤i≤k.
WerecallthataneffectivedivisorD⊂S iscalled numericallym-connectedif,when- ever D =D1+D2 withD1 and D2 effective and non-zero,one hasD1·D2 ≥m. The following resultisusedinSections 4and 5.
Theorem 1.6. (See[18,17].)LetS andL beas inTheorem 1.5 andassume c1(L)2>0.
Then, |L| has no fixed components if and only if every divisor in |L| is numerically 2-connected.
2. GeneralizedLazarsfeld–Mukaibundles
We startbyrecallingthedefinitionof LMbundles. We referto[1]foran exhaustive survey about the topic.If (A,V) is abase point freegdr on asmoothirreducible curve
C⊂S,theLMbundleEC,(A,V) isbydefinitionthedual ofthekerneloftheevaluation mapV ⊗ OSA.In particular,EC,(A,V) sitsintheshortexactsequence
0→V∨⊗ OS →EC,(A,V)→ωC⊗A∨→0,
andV∨ definesan(r+ 1)-dimensionalsubspaceoftheglobalsectionsofEC,(A,V).When thegrdonC iscomplete,thentheLMbundleissimplydenotedbyEC,A.Herearesome basicfactsconcerningLMbundles:
Proposition2.1. The LMbundlesEC,(A,V) satisfiesthefollowingproperties:
• rkEC,(A,V)=r+ 1,c1(EC,(A,V))=c1(L),c2(EC,(A,V))=d;
• χ(EC,(A,V))=g−d+ 2r+ 1,h1(EC,(A,V))=h0(A)−r−1,h2(EC,(A,V))= 0;
• EC,(A,V)is globallygenerated offthebaselocus ofωC⊗A∨;
• χ(EC,(A,V∨ ) ⊗EC,(A,V)) = 2(1−ρ(g,r,d)), and hence EC,(A,V) is non-simple if ρ(g,r,d)<0;
• if C ∈ |L| isgeneral and μ0,(A,V):V ⊗H0(ωC⊗A∨)→H0(ωC)is thePetri map, then thesimplicityof EC,(A,V) isequivalent totheinjectivityof μ0,(A,V).
In the sequel, we will especially treat the cases where ωC⊗A∨ is base point free.
We will show that everynon-simple globally generated LM bundle is theextension of twosheavesofaspecialtype.Withthis inmind,we introducethefollowing:
Definition 1.A torsion freesheaf E ∈ Coh(S) is called ageneralized Lazarsfeld–Mukai bundle(g.LMbundleinthesequel) iffh2(E)= 0 andeither
(I) E islocally freeandgeneratedbyglobalsectionsoffafiniteset;
or
(II) E isgloballygenerated.
Remark1. If conditions(I) and (II) of the abovedefinition are both satisfied, then E is the LM bundle associated with a smooth irreducible curve C ⊂ S and aprimitive linearseries (A,V) on C (i.e., E =EC,(A,V)). Furthermore, V = H0(A) if and only if h1(E)= 0.
Definition 1is motivatedfrom thefactthatg.LMbundleshavepropertiessimilarto LMbundles.Thisis provedinwhatfollows.
Proposition 2.2. LetE be a g.LM bundleof type (I)and rankrE,and Dt−1(V)denote the degeneracy locus of theevaluation map evV :V ⊗ OS → E for a general subspace V ∈G(t,H0(E)).Then,thefollowingare satisfied:
(i) ift≤rE−2,then Dt−1(V)isempty;
(ii) for t =rE−1, thelocus DrE−2(V) consists of a finite number of distinct points, none of whichlies inthelocuswhere E isnotglobally generated;
(iii) for t = rE, the locus DrE−1(V) is 1-dimensional. If c1(E)2 >0, then DrE−1(V) is an integral curve X (possibly singular at the points at which E is not globally generated) and the cokernel of the evaluation map evV is a torsion free sheaf of rank1on X.
Furthermore,givenanyclosedsubsetKofthetotalspaceofEsatisfyingthecondition dimK=rE−t+ 1,theimage of evV meets K ina(possiblyempty) finite set.
Proof. Welookattheshort exactsequence
0→E˜→E→τ →0, (1)
where E˜ is aglobally generated subsheaf of E satisfying H0( ˜E) H0(E), and τ is a 0-dimensionalsheafsupportedonthepointsatwhichEisnotgloballygenerated.Since h1( ˜E)=h1(E)+l(τ),thereexistsanexactsequence
0→V1⊗ OS ev1
−→E−→p E→τ →0, (2) where dimV1 =l(τ) and the cokernel of ev1 coincides with E.˜ It canbe easily shown thatE isagloballygenerated vectorbundle onS satisfyingh0(E)=h0(E)+l(τ) and hi(E)=hi(E) fori= 1,2 (cf.[10,Lemma 1.6]).
Take a general V2 ∈ G(t,H0(E)). By a general position argument (cf. [8, B.9.1]), if t ≤ rE −2 (respectively t = rE −1) one can choose V2 such that the degeneracy locus of the evaluation map ev = (ev1,ev2) : (V1 ⊕V2)⊗ OS → E coincides with that of ev1 (resp.is 0-dimensional and for allx ∈ supp(τ) thekernel of evx coincides with that of (ev1)x). Thus, (i) and (ii) follow because the map induced by p on the globalsectionssendsageneralV2∈G(t,H0(E)) toageneralpointoftheGrassmannian G(t,H0(E)).
Given ageneralsubspaceV2∈G(rE−1,H0(E)),we consider thefollowingcommu- tativediagram:
0 0
0 V1⊗ OS
ev1
E detE⊗Iξ τ 0
0 V1⊗ OS ev1
E E τ 0.
V2⊗ OS V2⊗ OS
0 0
ThesheafEislocallyfreeandξisa0-dimensionalsubschemeofSdisjointfromsupp(τ).
The cokernel E˜ of ev1 is aglobally generated, torsion free sheaf of rank 1 on S. The vanishing locus X of ageneral section α ∈ H0( ˜E) is 1-dimensional; if the inequality c1(E)2 > 0 is satisfied, then X is an integral curve, which is smooth whenever τ is curvilinear.Onecanliftα toasectionα∈H0(E).Having denotedbyV theimage of V2,αinH0(E),theevaluationmap evV isinjectiveanditscokernelisapuresheafof dimension 1 supportedonX.Hence,item(iii)follows.
Concerning the last part of the statement, one has dimp−1(K) ≤ dimK +l(τ).
IfV2 ∈G(t,H0(E)) is general,then [8,B.9.1] impliesthatthe image ofthe evaluation mapev2:V2⊗OS →Emeetsp−1(K) inatmostafinitenumberofpoints;thisconcludes theproof. 2
As regardsg.LM bundles of type (II), thefollowing analogue of Proposition 2.2(iii) holds;we onlygiveasketchoftheproofsincewedonotuseitintherestofthepaper.
Proposition 2.3. LetE be a g.LM bundleof type (II) and rank rE, and take ageneral subspace V ∈G(rE, H0(E)). Then, theevaluation mapevV :V ⊗ OS →E is injective anditscokernelisapuresheafB ofdimension1on S.If moreoverc1(E)2>0,then B isatorsionfree sheafof rank1onan integralcurve X⊂S.
Proof. Weconsidertheshort exactsequence
0→E →E∨∨→κ→0, (3)
whereκis a0-dimensional sheafonS.It isenoughto observethatH0(E)⊂H0(E∨∨) generatesE∨∨ offthesupportofκandto applyBertini’sTheorem. 2
Ournextgoalisto findalowerboundforthesecond Chernclassofag.LMbundles ofgivenrank.It isconvenientto introducethefollowing:
Definition2.LetE beag.LMbundle. TheClifford indexof E is:
Cliff(E) :=c2(E)−2(rkE−1).
Remark2.IfE =EC,A forasmoothirreducible curveC⊂S andA∈Pic(C),onehas theequalityCliff(EC,A)= Cliff(A).
Startingfrom thisobservation,we provethefollowing:
Proposition2.4. LetE be ag.LM bundlesuchthat c1(E)2 >0.If E isof type (I), then thefollowinginequalityissatisfied:
Cliff(E)≥2h1(E) +l(τ),
where τ is the0-dimensionalsheaf appearing in theexact sequence (1).If instead E is of type (II),we have:
Cliff(E)≥Cliff E∨∨
+l(κ),
where κisthe0-dimensionalsheaf appearing intheexact sequence(3).
Proof. Firstofall,assumeEsatisfiesbothconditions (I)and (II)inDefinition 1.Then, there exist asmooth irreduciblecurve C⊂S and A∈Pic(C) suchthatEC,A sitsina short exactsequence:
0→ OS⊕h1(E)→EC,A→E→0. (4) It turnsoutthat
Cliff(E) = Cliff(EC,A) + 2h1(E)≥2h1(E), (5) where thelastinequalityfollowsfromClifford’sTheorem.
Now,we considerthecasewhereEisoftype (I).Thegloballygeneratedvectorbundle Eappearingintheexactsequence(2)satisfiesh1(E)=h1(E),andc2(E)=c2(E)+l(τ), and rkE= rkE+l(τ).Inequality(5) forE yields:
Cliff(E) = Cliff(E) +l(τ)≥2h1(E) +l(τ). (6) Inorder to coverthecaseofag.LMbundleE oftype (II),it is enoughtoremark that rkE∨∨= rkE andc2(E∨∨)=c2(E)−l(κ). 2
Corollary2.5. LetE be ag.LM bundle of rankrE andc1(E)2>0.Then, Cliff(E)≥0 and equalityholds onlyin thefollowingcases:
(a) rE= 1 andE isagloballygenerated linebundle;
(b) E=EC,ωC forsomesmoothirreducible curveC⊂S ofgenus g=rE ≥2;
(c) rE >1 and E =EC,(rE−1)g21 for some smooth hyperelliptic curve C ⊂S of genus g > rE.
Proof. Proposition 2.4 triviallyimpliesthe firstpartof thestatement. If Cliff(E)= 0, then E is both locallyfree andglobally generated andsatisfies h1(E)= 0. Hence,E is the LM bundleassociated with asmooth irreducible curveC ⊂S and aprimitive line bundleA∈Pic(C) suchthatCliff(A)= 0.Case (a)occurswhenAisthestructuresheaf ofC.If insteaddeg(A)>0,thenClifford’sTheoremimpliesthatAiseitherthecanonical sheaf ωC (case (b)), or a multipleof a linear series of type g12 on C (case (c)). In the latter case, the inequality g > rE follows from the factthat the residual of the linear
series of type(rE−1)g12 is globally generated (it is aquotient of E)and by imposing (rE−1)g21=ωC. 2
Withregard to item (c)of Corollary 2.5, we recall theclassification ofhyperelliptic linearsystemsonS dueto Saint-Donat[18,Theorem 5.2].
Theorem 2.6. Let C ⊂ S be a smooth hyperelliptic curve of genus g ≥ 2 and define L:=OS(C).Then, oneof thefollowingoccurs:
• The equalityc1(L)2= 2 holds.
• There isasmooth,irreducible curve B⊂S of genus 2satisfying L OS(2B).
• There existsanirreducible ellipticcurveΣ⊂S suchthat c1(L)·Σ= 2.
Heretofore, we have treated g.LM bundles E satisfying c1(E)2 > 0. The following resultconcernsg.LMbundles whosefirstChernclasshasvanishingself-intersection.
Proposition 2.7. LetE be a g.LM bundlesuchthat c1(E)2 = 0.Then, E is both locally free and globally generated and satisfies c2(E) = 0. Furthermore, if h1(E) = 0, then E=OS(Σ)⊕rkE foran irreducible ellipticcurveΣ⊂S.
Proof. LetE be oftype (I). Uptoreplacing E with thebundle E sittingintheshort exactsequence
0→ O⊕Sh1(E)→E →E→0, (7)
we canassume h1(E)= 0. Then, Proposition (1.5) in[10] yields E = OS(Σ)⊕rkE for anirreducibleellipticcurveΣ⊂S andourstatementfollows trivially.
Now,letE beoftype (II)andconsider theshort exactsequence 0→E →E∨∨→κ→0.
SinceE∨∨ is ag.LMbundle oftype (I), it satisfiesc2(E∨∨)= 0.Therefore, in order toprovethatκ= 0,it is enoughto showthatc2(E)= 0, too.To thispurpose wemay assumeh1(E)= 0.We defineD(E) tobe thedual ofthe kernelof theevaluationmap H0(E)⊗ OS E,andwe obtainthefollowingexactsequence:
0→E∨→H0(E)∨⊗ OS →D(E)→ Ext1(E,OS)→0.
The condition h1(E) = 0 yields the isomorphism H0(D(E)) H0(E)∨. As a con- sequence, D(E) is aLM bundle oftype (I) which isglobally generated off thesupport ofExt1(E,OS) Ext2(κ,OS).Sincec1(D(E))2 = 0,we concludethatExt2(κ,OS)= 0 andκ= 0 aswell. 2
Remark 3. Proposition 2.7 trivially implies that, if E is a g.LM bundle satisfying c1(E)2= 0, thenCliff(E)=−2(rkE−1).
3. Coherentsystems
Wegivesomebackgroundinformationaboutcoherentsystemsandreferto[15,11]for details.
Definition 3.A coherentsystem of dimension d ona smooth projective variety Y is a pair (E,V),where E ∈Coh(Y) is ad-dimensionalsheaf and Vis avector subspaceof H0(E).A morphismofcoherentsystemsf : (E,V)→(E,V) isamorphismofcoherent sheaves f :E→E suchthatf(V)⊂V.
GivenacoherentsystemΛ= (E,V),we denotebyExti(Λ,−) thederivedfunctorsof thefunctorHom(Λ,−).We recallthefollowingresult:
Proposition 3.1. (See[11,Corollaire 1.6].)Given twocoherent systemsΛ= (E,V) and Λ= (E,V)on Y,thereexistsan exact sequence:
0→Hom Λ, Λ
→Hom E, E
→Hom V, H0
E /V
→Ext1 Λ, Λ
→Ext1 E, E
→Hom V, H1
E
→Ext2 Λ, Λ
→Ext2 E, E
→Hom V, H2
E
→ · · ·.
Inparticular, ifE isnon-simple, thenthecoherentsystem(E,H0(E)) isnon-simple as well.
Onecanintroduceanotionofstabilityforcoherentsystemsdependingonthechoice of a polynomialα∈ Q[t] with positiveleading coefficient. Having fixed α, we define a polynomialpα(E,V)∈Q[t] bysetting
pα(E,V)(t) := dimV rE
α(t) +pE(t),
where pE isthereducedHilbertpolynomialofE andrE isitsmultiplicity.We saythat (E,V) isα-semistableifE ispureandany coherentsubsheafF ⊂Esatisfies:
pα(F,V)≤pα(E,V), withV :=V ∩H0(F). (8) Theα-stability for(E,V) requires thatinequality(8)be alwaysstrictand impliessim- plicityof(E,V).
Wespecialize tothecaseofasmoothprojective K3 surfaceS and fixαequalto the Hilbert polynomialpS of OS; for convenience, we will talk about (semi)stability when
referringtopS-(semi)stability.Withthischoice,inequality(8)becomesequivalenttothe requirement:
dimV
rF ≤dimV rE
and, if “=” holds, thenpF(t)≤pE(t). (9) Remark4.If(E,V) isasemistablecoherentsystemofdimensiond,thentheevaluation mapevV :V ⊗ OS →Eis genericallysurjective. In particular,ifE hasnotorsion (i.e., d= 2),we obtainthatrk(E)≤dimV.
Givenan unstablecoherentsystem(E,V) suchthatE is torsionfree,we lookat its Harder–Narasimhanfiltration
0⊂
HN1(E), V1
⊂. . .⊂
HNs−1(E), Vs−1
⊂
HNs(E), Vs
= (E, V).
Weusethefollowing:
Lemma 3.2. Assume that E is globally generated by the sections in V and h2(E)= 0.
Thenforallithefollowingare satisfied:
(1) Vi=V ∩H0(HNi(E)).
(2) The sheaf HNi(E)isgenerically generatedby thesectionsin Vi.
(3) Forany subsheaf F ⊆HNi(E)with H0(F)∩V Vi,onehas h2(F)= 0.
The same holds true if (E,V) is semistable and we replace the Harder–Narasimhan filtrationwith theJordan–Hölderone.
Proof. First of all, note thatthe vanishing h2(E) = 0 implies dimV > rkE; indeed, if wehaddimV = rkE,thenEwouldbeisomorphictothedirectsumofrkE copiesof OS and h2(E)>0.
Since for all i the pair (E/HNi(E),V /Vi) satisfies the same properties as (E,V), by induction on the length of the filtration it is enough to prove the lemma for the maximal destabilizing coherent system (HN1(E),V1), which is semistable. Then, the firsttwopointstriviallyfollowfrom maximalityand Remark 4.
LetF beas inpoint (3);then, onehasrkF = rkHN1(E)<dimV1 and thesheafT appearingintheshortexactsequence
0→F →HN1(E)→T →0
is a torsion sheaf. By Serre duality, one has to show that Hom(F,OS) = 0. Assume there exists anon-zeromorphism ψ:F → OS and denote its kernelby K.Since Imψ istorsionfreeandgenericallygeneratedbyglobal sections,thenψissurjective andthe pair(K,H0(K)∩V1) destabilizes(HN1(E),V1),thusyielding acontradiction. 2
Proposition3.3.LetEbeavectorbundleonSgenericallygeneratedbyitsglobalsections.
Assume moreover that every subsheaf F ⊆ E for which H0(F) H0(E) also satisfies H2(F)= 0.
Then, eitherthe1-dimensionallocuswhereE isnotgeneratedbyitsglobalsectionsis a (possiblyempty)union of (−2)-curves, or thereexistsanef linebundleN along with an injectivemapN →E.
Proof. Thereexists ashort exactsequence
0→E˜→E→T →0, (10)
where T isapuresheafofdimension 1 supportedona(possiblysingularandreducible) curve γ, andthesheafE˜ satisfiesH0( ˜E) H0(E) andisglobally generatedoffafinite set.Since weareonasurface,thesheaf T is reflexive(cf.[12,Proposition 1.1.10]),i.e., T T∨∨ withT∨:=Ext1(T,OS).From(10)onegetsthatExti( ˜E,OS)= 0 fori= 1,2, hence E˜ is locally free. By assumption, h2( ˜E) = 0; therefore, E˜ is a g.LM bundle of type (I).Letγ=γ1∪ · · · ∪γsbe thedecompositionofγ inirreducible componentsand letri denotethegenericrankofT alongγi.
First of all, we reduce to the case s = 1. Define γi := γ1∪. . .∪γˆi∪. . . γs for all 1 ≤ i ≤ s. The restriction of T to γi may have some torsion T0(T|γi) (cf. [2, 3.1] for a detailed discussion); set T(i) := T|γi/T0(T|γi) and denote by T(i) the kernel of the surjectivemap T →T(i),whichissupportedonγi.Considerthefollowingcommutative diagram:
0
T(i) 0
0 E(i) E T(i) 0
0 E˜ E T 0.
0 T(i)
0
Since T(i) is pure, then E(i) is locally free and T(i) is pure as well. The1-dimensional locus at whichE(i) isnotglobally generatedcoincides with γi and H0(E(i)) H0(E).
Therefore, it isenoughto treatthecasewhereγ isirreducible andprovide aninjective morphism OS(γ)→E.
Lets= 1.Therestrictionof(10)toγ=γ1 givesanexactsequence:
0→K→E˜|γ →E|γ →T →0.
WelookatKasasubsetofthetotalspaceofE˜withdimK=r1+1.ByProposition 2.2, givenageneralsubspaceV ∈G(rkE−r1,H0( ˜E)),theimageoftheevaluationmapevV
meets K on a reduced 0-dimensional subscheme ζ ⊂ γ, that we may assume disjoint fromthesingularlocusofγ.We lookatthediagram:
0 0
0 E˜ E α T 0
0 E˜ E T 0.
V ⊗ OS V ⊗ OS
0 0
If r1 = 1, one can choose V such that E˜ = det ˜E ⊗Iξ and E = detE ⊗Iξ ⊗Iζ, whereξ isa0-dimensionalsubschemeofS disjoint fromγ.If insteadr1>1,thenE˜ is locallyfreewhileE istorsionfreeandsatisfiesSingE=ζ.Themapαfactorsthrougha surjectivemapα :E|γ →T.SinceE|γ hastorsionalongζandhasthesamerankasT, we obtaindet ˜E= detE(−γ) ifr1= 1,and E˜=E(−γ)∨∨ ifr1>1.Theisomorphism H0( ˜E) H0(E) implies that every section of E vanishes along γ and thus factors throughamap OS(γ)→ι E.Theirreducibilityofγyieldsthateitherγ isa(−2)-curve, orOS(γ) isnef.In bothcasesthevanishing h1(OS(γ))= 0,givenbythestrongversion ofBertini’sTheorem,enablesustoliftιtoamorphismOS(γ)→E.Thisconcludesthe proof. 2
IfE isavectorbundleas inProposition 3.3, we saythatE is anelementarymodifi- cationofag.LM bundleof type (I).As acorollary,we obtainthefollowing:
Theorem 3.4. Let EC,A be a non-simple LM bundle on S associated with a primitive line bundle A on C. Then, E is the extension of a g.LM bundle E2 of type (II) by a vector bundle E1, which is an elementary modification of a g.LM bundle of type (I);
in particular,one has h1(E2)=h2(E1)= 0.
Proof. Since EC,A is non-simple, the coherentsystem (EC,A,H0(EC,A)) is not stable.
We look atitsmaximaldestabilizingsequence:
0→
E1, H0(E1)
→
EC,A, H0(EC,A)
→(E2, V2)→0, (11)
The sheaf E2 is torsion free and, being a quotient of E, it is globally generated by V2 and satisfies h2(E2) = 0, that is, E2 is a g.LM bundle of type (II). Concerning the pair(E1,H0(E1)),the statementfollows directlyfrom Lemma 3.2 and short exact sequence (10)forE1. 2
Remark 5. In the next section it will be convenient to replace the maximal destabi- lizing sequence (11) with a minimal destabilizing sequence, where we require (E2,V2) to be a stable quotient of (EC,A,H0(EC,A)) with p(E2,V2) minimal. In other words, the coherent system (E2,V2) is any stable quotient of the minimal destabilizing quo- tient(EC.A/HNs−1(E),H0(EC,A)/H0(HNs−1)).By construction,minimaldestabilizing sequences are not unique. By Lemma 3.2 and short exact sequence (10) for E1, the subbundleE1appearinginsuchaminimaldestabilizingsequenceisanelementarymod- ificationofag.LMbundle oftype (I)and hencesatisfies h2(E1)= 0.
4. Linearseriescomputingthe Cliffordindex
In this section, we describe complete linear series A of type gdr on a curve C ⊂ S such thatCliff(C)= Cliff(A) and ρ(g,r,d)<0.The following lemma explains how to recognizewhenalinearseriesonCiscutoutbytherestrictionofalinebundleonS by onlylookingatthecorresponding LMbundle.
Lemma 4.1. LetN1 ∈Pic(S) satisfy h0(N1)≥2andh1(N1)= 0.Also assume that the linebundleN2:=L⊗N1∨isgloballygeneratedandh1(N2)= 0.LetE beag.LMbundle on S.Then, E=EC,N2⊗OC forsome smoothirreduciblecurveC∈ |L|if andonly ifE is anextensionof theform:
0→N1→E→EC2,ωC2 →0, (12)
forsome smoothirreducible curveC2∈ |N2|.
Proof. First,assumeE=EC,N2⊗OC forasmoothirreduciblecurveC∈ |L|.SinceN2is globally generatedandh1(N1)= 0,we obtainadiagram:
0
N1∨ 0
0 E∨C,N2⊗OC H0(N2⊗ OC)⊗ OS N2⊗ OC 0
0 K∨ H0(N2)⊗ OS
N2 0.
0 N1∨
0
Onecaneasily checkthatK isagloballygeneratedvector bundlesatisfyinghi(K)= 0 for i = 1,2, that is, K is a LM bundle. Furthermore, one has c1(K) = c1(N2) and c2(K) = c1(N2)2 = 2(rkK −1). Corollary 2.5 thus yields K EC2,ωC2 for a smooth irreducible curve C2 ∈ |N2| and the “only if” part of the statement fol- lows.
Asfortheconverseimplication,letE sitinashort exactsequenceas in(12).Then, E is locally freeandglobally generated,that is, E=EC,A for asmoothcurve C∈ |L| andalinebundleA∈Picd(C) withd=c2(E).Lookatthediagram:
0 H0(C, A)∨⊗ OS E α ωC⊗A∨ 0.
N1
ι
Since h0(N1)≥2,then Hom(N1,OS)= 0 and 0=α◦ι∈Hom(N1,ωC⊗A∨). By ad- junction, we get h0(A∨⊗N2⊗ OC) >0. It is easy to verifythat deg(N2⊗ OC)= d, henceAisisomorphictoN2⊗ OC. 2
Remark6. The aboveproof also shows that,as soon as there exists N ∈ Pic(S) such that h0(N) ≥ 2 together with an injective morphism N → EC,A, one obtains that h0(A∨⊗(L⊗N∨)⊗ OC)= 0,i.e.,thelinearseries|A|iscontainedin|(L⊗N∨)⊗ OC|; thiscoincideswith therestrictionof|L⊗N∨|toC ifh1(N)= 0.
Example 1. Assumethe existence of an irreducible curve B ⊂S of genus 2 such that c1(L)= 2B.ForanysmoothirreduciblecurveC∈ |L|,thecompletelinearseriesOC(B)
