COHOMOLOGICAL SUPPORT LOCI FOR ABEL-PRYM CURVES

18 Pages·2008·0.17 MB·English

LEMATEMATICHE
Vol.LXIII(2008)–Fasc.I,pp.205–222
COHOMOLOGICALSUPPORTLOCIFOR
ABEL-PRYMCURVES
S.CASALAINAMARTIN-M.LAHOZ-F.VIVIANI
ForanAbel-PrymcurvecontainedinaPrymvariety,wedeterminethe
cohomologicalsupportlociofitstwistedidealsheavesandthedimension
ofitstheta-dual.
Introduction
Thepurposeofthispaperistostudythetheta-dualandthecohomologicalsup-
port loci for the twisted ideal sheaves of an Abel-Prym curve contained in the
Prym variety associated to an e´tale double cover of smooth projective non-
hyperellipticcurves.
Recall that given a coherent sheaf F on a smooth projective variety X, the
i-thcohomologicalsupportlocusofF is
Vi(F):={α ∈Pic0(X)|hi(X,F⊗α)>0}⊂Pic0(X).
These loci have been studied in a number of contexts, and were considered for
example by Green-Lazarsfeld (see [5, 6]) in order to prove a generic vanish-
Entratoinredazione22febbraio2008
AMS2000SubjectClassification:14H40,14K12,14F17
Keywords: Prym varieties, Abel-Prym curves, cohomological support loci, generic vanishing,
theta-dual
ThefirstauthorwaspartiallysupportedbyNSFMSPRFgrantDMS-0503228.Thesecondauthor
hasbeensupportedbyMinisteriodeEducacio´nyCiencia,becadeFormacio´ndelProfesorado
Universitario,MTM2006-14234-C02and2005SGR-00557.
206 SEBASTIANCASALAINAMARTIN-MART´ILAHOZ-FILIPPOVIVIANI
ing theorem for the canonical sheaf of irregular complex varieties. More pre-
\
cisely, they proved that if the Albanese morphism a:X →Alb(X)∼=Pic0(X)
hasgenericfiberofdimensionk,then
codim Vi(ω )≥i−k.
Pic0(X) X
In a sequence of articles ([11, 17, 18] cf. [16])), G. Pareschi and M. Popa
have studied similar questions, and have introduced the notion of a GV sheaf
k
(seealso[7]):
Definition. AcoherentsheafF issaidtobeGV forsomek∈Z(whichstands
k
forgenericvanishingoforderk)if
codim Vi(F)≥i−k foralli>0. (*)
Pic0(X)
ObservethatwehavethenaturalinclusionsbetweentheGV -sheaves:
k
(W)IT =GV ⊂···⊂GV =M⊂GV =GV ⊂···⊂GV =Coh(F)1,
0 −d −1 0 d
where the sequence becomes stationary outside the interval [−d,d] for d =
dim(X). With this terminology, the result of M. Green and R. Lazarsfeld says
thatiftheAlbanesemorphismhasgenericfiberofdimensionk,thenω isGV .
X k
The above condition (∗) can be expressed in terms of the Fourier-Mukai
transform with respect to the kernel P =(a,id)∗(L)∈D(X×Aˆ), where L
is the Poincare´ line bundle on A×Aˆ (suitably normalized) and A = Alb(X).
G. Pareschi and M. Popa use this to study vanishing in a variety of contexts,
and in particular when F is an adjoint linear series or an ideal sheaf suitably
twisted;thishasproducedmanyinterestingapplications(see[12–14,17,18]).
When X = A is an abelian variety, the case we will be considering here,
Pic0(A) is canonically isomorphic to Ab, and soVi(F) can be considered as a
subspace inside the dual abelian variety. Moreover, in the case of a principally
polarizedabelianvariety(A,Θ),whichwewillabbreviatebyppavinthesequel,
wecanviewthecohomologicalsupportlociassubspacesoftheoriginalabelian
varietyusingtheisomorphismϕ :A→∼ Aˆ. Inthissetting,Pareschi-Popaintro-
Θ
duced the following definition of theta-dual (see [18, Def. 4.2]), which can be
viewedasthecohomologicalsupportlocusofatwistedidealsheaf.
1In the above chains of inclusions we have also indicated some other names that are used
intheliterature, namely: thesheavessatisfyingGV arealsocalledGV-sheaves(whichstands
0
for generic vanishing sheaves), the GV−1-sheaves are called M-regular sheaves (which stands
Mukai regular sheaves) and the GV -sheaves are called (W)IT -sheaves because they satisfy
−d 0
Vi(F)=0/ foralli>0,orinotherwordstheysatisfiesthe(weak)indextheoremwithindex0in
Mukai’sterminology.
COHOMOLOGICALSUPPORTLOCIFORABEL-PRYMCURVES 207
Definition. Given a closed subscheme X of a ppav (A,Θ), the dual theta-dual
V(X)ofX istheclosedsubset2 definedby:
V(X):=V0(IX(Θ))={α ∈Aˆ |h0(A,IX(Θ)⊗α)6=0}⊂Ab∼=Pic0(A).
Observethat,viatheprincipalpolarizationΘ,thetheta-dualV(X)canbecanon-
icallyidentifiedwiththelocus{a∈A|X ⊂Θ =t∗Θ}oftheta-translatescon-
a a
tainingX (heret :A→Aisthetranslationmap).
a
More generally, we will be interested in the cohomological support loci of
the formVi(I (nΘ)). In particular, in this paper we consider the cohomolog-
X
ical support loci associated to the ideal sheaves of Abel-Prym curves. To fix
some notation, let P be the Prym variety of dimension g−1 associated to the
e´tale double cover Ce→C of irreducible smooth projective non-hyperelliptic
curvesofgenusg˜=2g−1andg≥3,respectively. LetΞbethecanonicalprin-
cipal polarization. SinceCeis not hyperelliptic, there is an embeddingCe,→P,
unique up to translation. We denote by I the ideal sheaf ofCeinside P (see
Ce
section1forthedefinitionandthestandardnotations). Ourmainresultis(com-
biningTheorems2.2,3.1,4.2):
Theorem A. Let Ce be a non-hyperelliptic Abel-Prym curve embedded in its
Prymvariety(P,Ξ). Thenwehave:
1. Thetheta-dualofCehasdimension
dim(V(Ce))=dimP−3=g−4.
2. ThecohomologicalsupportlociofI (Ξ)canbe(non-canonically)
Ce
identifiedwith
V0(I (Ξ))=V1(I (Ξ))=V(Ce),
 Ce Ce
V2(I (Ξ))=P,
Ce

V≥3(I (Ξ))=0/.
Ce
3. ThecohomologicalsupportlociofI (2Ξ)canbe(non-canonically)
Ce
identifiedwith
 (
P ifg≥4,
V0(I (2Ξ))=
 Ce {0} ifg=3,
V1(I (2Ξ))=V2(I (2Ξ))={0},
 Ce Ce

V≥3(I (2Ξ))=0/.
Ce
2It is possible to put a canonical schematic structure onV(X) (see [18, Def. 4.2]), which
howeverwillneverplayaroleinthispaper.
208 SEBASTIANCASALAINAMARTIN-MART´ILAHOZ-FILIPPOVIVIANI
The above theorem follows by combining Theorems 2.2, 3.1, 4.2 together
with the identifications that we make in formulas (1) and (2) below. As an
immediateCorollaryweobtain:
CorollaryB. WiththenotationoftheaboveTheorem,thefollowinghold
(i) I (Ξ)isnotGV. Moreprecisely,itisGV butnotGV .
Ce 2 1
(ii) I (2Ξ)isGV,butnotIT . Moreprecisely,itisGV butnot
Ce 0 −(g−2)
GV .
−(g−1)
(iii) I (mΞ)isIT foreverym≥3.
Ce 0
Statements (i) and (ii) follow by direct inspection from the theorem, while
part(iii)followsfrompart(ii)andthegeneralfactthatifF isaGV-sheafthen
F(Θ)isIT (see[18,Lemma3.1]).
0
These results should be compared to the case of an Abel-Jacobi curve.
Recall that for any non-rational curve C, the Abel-Jacobi map gives a non-
canonical embedding of C in its canonically polarized Jacobian (J(C),Θ). In
this case, the cohomological support loci can be (non-canonically) identified
with(combine[12,Thm. 4.1,Prop. 4.4]and[18,Lem. 3.3,Exa. 4.5])
V(C)=V0(I (Θ))=V1(I (Θ))=W ,
 C C g−2

V0(I (nΘ))=J(C)forn≥2,
C

V≥2(I (Θ))=V≥1(I (nΘ))=0/ forn≥2,
C C
whereW =W0 is the Brill-Noether locus of line bundles of degree g−2
g−2 g−2
withnon-trivialglobalsections. Fromtheabovedescription,wegetthatforan
Abel-Jacobicurve
(i) I (Θ)isGV,butnotIT .
C 0
(ii) I (mΘ)isIT foreverym≥2.
C 0
(iii) dim(V(C))=g−2=dimJC−2.
In[18],Pareschi-Popahaveprovedthattheabovecondition(i)characterizes
Abel-Jacobicurvesamongthenon-degeneratecurvesinsideappav. Moreover,
theyhavefurtherconjecturedthattheconditions(ii)and(iii)shouldalsoprovide
newcharacterizationsofAbel-Jacobicurves3.
3Wereferto[18]foranalogousconjecturesconcerningthesubvarietiesofappavofminimal
cohomologicalclass.
COHOMOLOGICALSUPPORTLOCIFORABEL-PRYMCURVES 209
The results in this paper show that this conjecture is not violated by Abel-
Prymcurves,whichinasensearethecurvesinsideappavclosesttotheAbel-
Jacobicurves.
Inaddition,fromtheresultsonAbel-Prymcurvesabove,andthosecitedfor
Abel-Jacobicurves,itseemsnaturaltoaskfortherelationbetweenthefollow-
ingconditionsonacurveX onappav(A,Θ)ofdimensiong:
(1) I (eΘ)isGV,butnotIT .
X 0
(2) I ((e+1)Θ)isIT ,butI (eΘ)isnot.
X 0 X
(3) dimV(X)=g−e−1.
(4) X isanAbel-Prym-TyurincurvewithPrym-Tyurinvariety(A,Θ)⊂(JX,
Θ )ofexponente,thatis[X]= e[Θ]g−1.
X (g−1)!
Prym-Tyurin varieties of exponent 1 are precisely the Jacobians (by the
Matsusaka-Ran criterion), so that the Pareschi-Popa conjecture states that the
aboveconditionsareallequivalentfor e=1. Inthenextcase, itisknown(see
[20]) that the closure inside the moduli space A of ppav’s of dimension g of
g
the Prym-Tyurin varieties of exponent 2 has a unique component of maximal
dimension (which is 3g), namely the closure of the classical Prym varieties4.
Therefore,ourresultsinTheoremAshowthat“most”oftheAbel-Prym-Tyurin
curvesofexponent2satisfiestheconditions(1),(2)and(3).
Ontheotherhand,AndreasHoeringhaspointedouttousthatcondition(3)
ismuchweakerthancondition(4): anycurveX onasubvarietyY withdim(Y)>
1anddimV(Y)=g−e−1willhavedimV(X)≥g−e−1. SinceY willcontain
curves of arbitrarily high degree with respect to Θ, one can construct curves
satisfying (3) but not (4). As a concrete example, consider a curve X lying
on a W (1 < d < e+1) inside a Jacobian or on the Fano surface inside the
d
intermediateJacobianofacubicthreefold. Thusweproposeanalternateversion
of(3),whichmaybemorecloselyrelatedtotheotherconditions
(30) dimV(X)=g−e−1 and X is not contained in a subvarietyY with 1<
dim(Y)<e+1anddimV(Y)=g−e−1.
SinceanAbel-PrymcurveoftheintermediateJacobianofacubicthreefold
lies on the Fano surface F, which has class [Θ]3/3! and dimV(F)=g−3, (4)
doesnotimply(30),andsowesuggestthefollowingmodificationof(4)aswell:
4AmongthePrym-Tyurinppavofdimensiongandexponent2,theclassicalPrymvarieties
canalsobecharacterizedasthoseforwhichthecurveX issmoothofmaximalarithmeticgenus,
namely2g+1.
210 SEBASTIANCASALAINAMARTIN-MART´ILAHOZ-FILIPPOVIVIANI
(40) X isanAbel-Prym-TyurincurveofexponenteandX isnotcontainedin
asubvarietyY with1<dim(Y)=d<e+1andclass[Y]=α[Θ]g−d with
(g−d)!
α <e.
The paper is organized as it follows. In section 2, we review the definition
of the Prym variety (P,Ξ) associated to an e´tale double coverCe→C in order
to fix the notation used throughout the paper. In section 3, we prove that the
theta-dual of an Abel-Prym curveCeinside P can be set-theoretically identified
with the Brill-Noether locus V2 defined in [19]. General results about these
Brill-Noether ([2, 4]) give the inequality dimV(Ce)≥dim(P)−3. Using ideas
from[10, sections6, 7],weshowthattheequalityholds(Theorem2.2), which
proves part (1) of the Main Theorem A. In section 4 and 5, we compute the
cohomologicalsupportlociforthetwistedidealsheavesI (Ξ)(Theorem3.1)
Ce
and I (2Ξ) (Theorem 4.2), proving explicitly parts (2) and (3) of the Main
Ce
TheoremA.
1. Notationandbasicdefinitions
Throughoutthispaper,weworkoveranalgebraicallyclosedfieldkofcharacter-
isticdifferentfrom2. ThebasicresultscitedhereareduetoMumford[9]. Let
π :Ce→C be an e´tale double cover of irreducible smooth projective curves of
genusg˜andg,respectively. BytheHurwitzformula,wegetthatg˜=2g−1. We
denotebyσ theinvolutiononCeassociatedtotheabovedoublecover. Consider
thenormmap
Nm:Pic(Ce) −→ Pic(C)
OCe(∑jrjpj) 7−→ OC(∑jrjπ(pj)).
Thekernelofthenormmaphastwoconnectedcomponents
kerNm=P∪P0⊂Pic0(Ce),
wherePisthecomponentcontainingtheidentityelementandis,bydefinition,
thePrymvarietyassociatedtothee´taledoublecoverπ. Theabovecomponents
PandP0 havethefollowingexplicitdescription
n o
P= O (D−σ(D))|D∈Div2N(Ce),N ≥0 ,
Ce
n o
P0= O (D−σ(D))|D∈Div2N+1(Ce),N ≥0 .
Ce
ItisoftenusefultoconsidertheinverseimageofthecanonicallinebundleofC
viathenormmap. Thisalsohastwoconnectedcomponents
Nm−1(ωC)=P+∪P−⊂Pic2g−2(Ce)=Picg˜−1(Ce),
COHOMOLOGICALSUPPORTLOCIFORABEL-PRYMCURVES 211
whichhavethefollowingexplicitdescription
P+=(cid:8)L∈Nm−1(ω )|h0(L)≡0 mod2(cid:9),
C
P−=(cid:8)L∈Nm−1(ω )|h0(L)≡1 mod2(cid:9).
C
TheabovevarietiesP0,P+ andP− areisomorphictothePrymvarietyPand,in
thiswork,wewillpassfrequentlyfromonetoanother.
ThereisaprincipalpolarizationΞ∈NS(P)inducedbytheprincipalpolar-
ization ΘCe∈NS(JCe). In fact, ΘCe|P =2Ξ. One of the primary motivations for
consideringP+ istheexistenceofacanonicallydefineddivisorΞ+ whoseclass
intheNeron-SeverigroupofPisΞ:
n o
Ξ+= L∈P+⊂Picg˜−1(Ce)|h0(L)>0 ⊂P+.
Ontheotherhand,thecanonicalAbel-Prymmapisdefinedas
i:Ce −→ P0
p 7−→ σ(p)−p.
If Ce is hyperelliptic then the image of Ce via the Abel-Prym map is a smooth
hyperellipticcurveDandthePrymvarietyPisisomorphictotheJacobianJ(D)
of D ([3, Cor. 12.5.7]). On the other hand, if C is hyperelliptic but Ce is not,
thenthePrymvarietyPistheproductoftwohyperellipticJacobians(see[10]).
Therefore, since we are mostly interested in the case of an irreducible non-
Jacobianppav,wewillassumethroughoutthispaperthatC isnothyperelliptic
(and in particular g≥3). Note that under this hypothesis, the Abel-Prym map
isanembedding([3,Cor. 12.5.6]).
SincetheAbel-PrymcurveCe⊂P0 andthepolarizationΞ+⊂P+ liecanon-
ically in different spaces, the cohomological support loci for the twisted ideal
sheafI (nΞ+)isonlydefineduptoatranslationsincewehavetochooseaway
Ce
to translateCeand Ξ+ inside the Prym variety P. For this reason, we introduce
thefollowingauxiliary(canonicallydefined)loci
Vei(I (nΞ+))={E ∈P−|hi(P0,I (nΞ+)>0}⊂P−, (1)
Ce Ce E
where I is the ideal sheaf of Ce inside P0 and for E ∈ P− ⊂ Picg˜−1(Ce), we
Ce
denote by Ξ+ ⊂P0 the translate of the canonical theta divisor Ξ+ by E−1. The
E
relation between Vei(I (nΞ+)) and Vi(I (nΞ)) is easy to work out. In fact
Ce Ce
thereisachoiceoftranslateofCe⊂PandΞ⊂Psothatundertheisomorphism
ψ : P− −→ P
E
0
E 7−→ E⊗E−1
0
212 SEBASTIANCASALAINAMARTIN-MART´ILAHOZ-FILIPPOVIVIANI
inducedbyalinebundleE ∈P−,wehave
0
Vi(I (nΞ))=Vei(I (nΞ+))⊗n⊗E⊗−n:={(E⊗E−1)⊗n|E ∈Vei(I (nΞ+))}.
Ce Ce 0 0 Ce
For this reason, we will also identify the cohomological support loci with
thefollowingcanonicalloci:
Vi(I (nΞ))=Vei(I (nΞ+))⊗n⊂Nm−1(ω⊗n)⊂Picn(g˜−1)(Ce). (2)
Ce Ce C
For later use, we end this section with the following Lemma, which de-
scribes the restriction of the translates of the theta-divisor to the Abel-Prym
curve.
Lemma1.1. GivenanyE ∈P−,thereisanisomorphismoflinebundles
OP0(Ξ+E)|Ce∼=E.
Moreover,ifE ∈P−−V(Ce)andDistheuniquedivisorin|E|,thenwehavean
equalityofdivisors
(cid:0)Ξ+(cid:1) =Ce∩Ξ+=D.
E |Ce E
Proof. This is standard, we include a proof for the convenience of the reader.
Suppose first that E ∈P−−V(Ce), which, by Lemma 2.1, is equivalent to the
conditionh0(Ce,E)=1. Write|E|=D=p1+...+pg˜−1,where pi∈Ce. Since pi
isafixedpointofthelinearseries|E|,wehavethath0(Ce,E⊗OCe(−pi+σpi))=
2,whichimpliesthat
D⊂Ce∩Ξ+=(cid:0)Ξ+(cid:1) .
E E |Ce
UsingthatCe·Ξ+=g˜−1,wegetthedesiredsecondequality. Nowconsiderthe
E
maps
Ce×P−(−a,→id)P0×P−−µ→P+,
where a is the Abel-Prym map and µ is the multiplication map. Let P be the
Poincare´ linebundleonCe×P−,trivializedoverthesection{p}×P− forsome
p∈Ce. ConsiderthelinebundleonCe×P−givenbyL :=(a×id)∗µ∗OP+(Ξ+).
WecantrivializeL alongthegivensection{p}×P−bytensoringwiththepull
backfromP−ofthedivisorΞ+ . ItiseasytocheckthatthefibersofP and
p−σ(p)
L overCe×{E}aregivenby
(P =E,
Ce×{E}
L =O (Ξ+) .
Ce×{E} P0 E |Ce
Bywhatwasprovedabove,ifE 6∈P−−V(Ce)thenthetwofibersagree. Bythe
Seesawtheorem(e.g. [3,Lemma11.3.4]),P ∼=L andwegetthedesiredfirst
equality.
COHOMOLOGICALSUPPORTLOCIFORABEL-PRYMCURVES 213
2. Thetheta-dualofCe
Inthissection, wewanttostudythetheta-dualofCeinthePrymvarietyP; this
canbeidentifiedcanonicallywiththeset(see(2)):
V(Ce):=(cid:8)E ∈P−|h0(P0,I (Ξ+))>0(cid:9)⊂P−.
Ce E
Infactthetheta-dualV(Ce)canbedescribedintermsofthefollowingstan-
dardBrill-Noetherloci(see[19]):
Vr :=(cid:8)L∈Nm−1(ω )|h0(L)≥r+1,h0(L)≡r+1 mod2(cid:9),
C
where Vr ⊂ P− (resp. Vr ⊂ P−) if r is even (resp. odd). We view both the
thetadualandtheBrill-Noetherlociassets,althoughtheycanbeendowedwith
naturalschemestructures5.
Lemma2.1. Wehavetheset-theoreticequality
V(Ce)=V2.
Proof. AnelementE∈P−belongstoV(Ce)ifandonlyifCe⊂Ξ+,which,bythe
E
definitionofCe⊂P0,isequivalenttoh0(Ce,E⊗O (σ(p)−p))>0forevery p∈
Ce
Ce. ByMumford’sparitytrick(see[10]),thishappensifandonlyifh0(Ce,E)≥3,
thatisE ∈V2.
Theorem2.2. Foranye´taledoublecoverCe→C asabovewithC nothyperel-
lipticofgenusg,itholdsthat
dim(V2)=dim(P)−3=g−4.
For g=3, the Theorem says thatV2 =0/. We start with the following Lemma,
whichissimilarto[10,Lemmap. 345].
Lemma2.3. IfZ⊆V2isanirreduciblecomponent,anddimZ≥g−3,thenfor
agenerallinebundleL∈Z,thereisalinebundleM onCwithh0(M)≥2,and
aneffectivedivisorF onCesuchthatL∼=π∗M⊗O (F).
Ce
Proof. LetZ andLbeasinthestatement. Supposethath0(L)=r+1forr≥2
even,sothatL∈Wr −Wr+1. Fromthehypothesis,wegetthat
g˜−1 g˜−1
dimT Wr ∩T P−≥g−3=dim(P−)−2. (3)
L g˜−1 L
5WeremarkhoweverthattheVrhavebeenconsideredwithdifferentnaturalschematicstruc-
tures(compare[19]and[4]).
214 SEBASTIANCASALAINAMARTIN-MART´ILAHOZ-FILIPPOVIVIANI
The Zariski tangent space toWr at L is given by the orthogonal complement
g˜−1
totheimageofthePetrimap(e.g. [1,Prop. 4.2]):
H0(Ce,L)⊗H0(Ce,σ∗(L))→H0(Ce,ω ),
Ce
where we have used that ω =π∗(ω )=L⊗σ∗(L). On the other hand, the
Ce C
tangentspacetothePrymisbydefinitionTLP−=H0(Ce,ωCe)−,the(−1)-eigen–
spaceofH0(Ce,ω )relativetotheinvolutionσ. Therefore,itiseasytoseethat
Ce
the intersection of the Zariski tangent spaces T Wr ∩T P− is given as the
L g˜−1 L
orthogonalcomplementtotheimageofthemap
v0:∧2H0(Ce,L)→H0(Ce,ωCe)−
definedbyv (s ∧s )=sσ∗s −s σ∗s.
0 i j i j j i
Theinequality(3)isequivalenttocodim(kerv )≤2. Ontheotherhand,the
0
decomposableformsin∧2H0(Ce,L)formasubvarietyofdimension2r−1≥3,
andsothereisadecomposablevectors ∧s inkerv . Thismeansthatsσ∗s −
i j 0 i j
s σ∗s = 0, or in other words that sj defines a rational function h in C. We
cojncluide by taking M =O ((h) ) asnid F the be the maximal common divisor
C 0
between(s) and(s ) .
i 0 j 0
ProofofTheorem2.2. ThedimensionofV(Ce)=V2 isatleastg−4bythethe-
orem of Bertram ([2], see also [4]). Suppose, by contradiction, that there is an
irreduciblecomponentZ⊆V2 suchthatdimZ=m≥g−3. Then,byapplying
theprecedingLemma2.3forthegeneralelementL∈Z,
L∼=π∗M⊗O (B)
Ce
where M is an invertible sheaf onC such that h0(M)≥2, and B is an effective
divisoronCesuchthat Nm(B)∈|KC⊗M⊗−2|. Thefamily ofsuchpairs(M,B)
isafinitecoverofthesetofpairs{M,F}where:
• M isaninvertiblesheafonCofdegreed≥2suchthath0(M)≥2,
• F is an effective divisor onC of degree 2g−2−2d ≥0, such that F ∈
|K ⊗M⊗−2|.
C
By Marten’s theorem applied to the non-hyperelliptic curve C (see [1, Pag.
192]),thedimensionoftheabovefamilyoflinebundlesMisboundedaboveby
dim(W1)<d−2. (4)
d
FixingalinebundleM asabove,thedimensionofpossibleF satisfyingthe
secondconditionisboundedbyClifford’stheorem,
h0(K ⊗M⊗−2)−1≤g−1−d, (5)
C