Green-Lazarsfeld's Conjecture for Generic Curves of Large Gonality

5 Pages·0.06 MB·English

by  Marian Aprodu

GREEN-LAZARSFELD’S CONJECTURE FOR GENERIC
CURVES OF LARGE GONALITY
3 Marian Aprodu∗ and ClaireVoisin
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2 February 1, 2008

n
a
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Abstract
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We use Green’s canonical syzygy conjecture for generic curves to prove that the
]
G Green-Lazarsfeld gonality conjecture holds for generic curves of genus g, and go-
A nalityd,ifg/3 < d< [g/2]+2.
.
h 1. Introduction
t
a
m Denoting by K (X,L) the Koszul cohomology with value in a line bundle L (see [5]),
p,q
[ Green andLazarfeld provedthefollowing(cf. Appendixto [5]):

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Theorem 1 LetX beacomplexmanifold,L ,andL betwolinebundlesonX suchthat
v 1 2
1 r1 := h0(L1)−1 ≥ 1,and r2 := h0(L2)−1 ≥ 1. Then Kr1+r2−1,1(X,L1 ⊗L2) 6= 0.
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2 LetnowC beasmoothcomplexsmoothprojectivecurveofgonality
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0 d := min{deg(L),h0(L) ≥ 2}.
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0
Green-Lazarsfeld’s theorem, applied to L = O (D), where deg(D) = d,
/ 1 C
h
h0(C,O (D)) = 2, and L = L−D withdeg(L) sufficientlylarge, implies
t C 2
a
m
Kh0(L)−d−1,1(C,L) 6= 0.
:
v
Thegonalityconjecturepredictsthatthisis optimal,namely:
i
X
r Conjecture 1 (Green and Lazarsfeld, [7]) For C a curve of gonality d, and for any line
a
bundleL ofsufficientlylargedegree, wehaveKh0(L)−d,1(C,L) = 0.
Inspiteoftheevidencebroughtbythe“K Theorem”of[5],whichsolvesthehyper-
p,1
ellipticcase(amongotherthings),afterhavingformulatedthegonalityconjecture,Green
and Lazarsfeld shown their mistrust of the statement they had just made. Since then, the
conjecture has been almost forgotten, and it took a while untill some new evidence was
discovered(see[3],[1]). Thisdelayisprobablyduetothefactthattheconjecturedidnot
count among the mathematical highlights of the last years, as almost all the attention in
thetheoryofsyzygiesofcurves wasfocused on themorefamousGreen conjecture.
The aim of this short note is to mix together the main results of [1] on the one hand,
and of [9], and [10] on the other hand, in order to verify the Green-Lazarfeld conjecture
forgenericcurves oflargegivengonality. Thefirst resultweproveisthefollowing:
∗SupportedbyanE.C.MarieCurieFellowship,contractnumberHPMF-CT-2000-00895
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Theorem 2 For any positive integers g and d such that g/3+ 1 ≤ d < [(g +3)/2], the
gonalityconjectureis validforgenericcurves ofgenusg andgonalityd.
Note that for generic curves of genus g the gonality equals [(g + 3)/2], and thus
Theorem 2coversall possible,not toosmall,gonalities,exceptforthegenericgonality.
Oursecond resultis:
Theorem 3 The gonalityconjectureisvalidfor genericcurves of evengenus.
Inthestatementsabove,thewordgenericshouldbereadintheusualsense. Thecom-
plexcurvesoffixedgenusg,andgonalityd,areparametrisedbyanirreduciblesubvariety
of the moduli space M , and a generic curve is a curve corresponding to a generic point
g
of this variety. Irreducibility follows from the well-known fact that the closure of this
subvarietyisactuallytheclosureoftheimageinM ofaHurwitzschemeandthenapply
g
[4].
Finally,wementionthatallthenotationweuseinthesequelisstandard,andwerefer
to [5]forbasicfacts about Koszulcohomology.
2. Proofs ofmainresults
First ofall,werecall thefollowingresultfrom [1]:
Theorem 4 IfLisanonspeciallinebundleonacurveC,whichsatisfiesK (C,L) = 0,
n,1
for a positive integer n, then, for any effective divisor E of degree e, we have
K (C,L+E) = 0.
n+e,1
Inparticular,ifKh0(L)−d,1(C,L) = 0foranonspeciallinebundleL,withh0(L)−d >
0, then Kh0(L′)−d,1(C,L′) = 0 for any L′ of sufficiently large degree. By means of the
Zariskisemi-continuityofgradedBettinumbers(see,forexample,[2]),forbothTheorem
2,andTheorem3,itsufficestoexhibitoned-gonalcurveC ofgenusg(whered = g/2+1
forTheorem 3), andonenonspeciallinebundleLonC satisfyingKh0(L)−d,1(C,L) = 0.
ProofofTheorem 2. Step1. ConstructionofC and L.
AsuitablechoiceofsuchacurveC isprovidedbytheproofofCorollary1of[9]. We
start with a K3 surface S whose Picard group is cyclic, generated by a line bundle L of
self-intersectionL2 = 4k−2,wherek = g−d+1. Wedenoteν = g−2d+2 ≥ 1. Under
these assumptions, as ν ≤ k/2, we know that there exists an irreducible curve X ∈ |L|,
havingexactlyν simplenodesassingularpoints,andnoothersingularities,andsuchthat
itsnormalizationC is ofgonalityd = k +1−ν, see[9].
WesetL = K (p+q),wherep,andq aretwodistinctpointsofC thatlieoveranode
C
x ofX.
Step 2. Recall themain resultof[9].
Theorem 5 The K3surfaceS being asabove, wehaveK (S,L) = 0.
k,1
It follows directly from this result, from the adjunction formula, and from the hyper-
plane section theorem for Koszul cohomology (see [5]) that K (X,K ) = 0. Since
k,1 X
k = h0(C,L)−d, theproofofourtheoremis concludedbythefollowinglemma.
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Lemma 1 Let X be a nodal curve, C −→ X be the normalization of X, and p,q ∈ C
be two distinct points lying over the same node x of X. Then, for any n ≥ 1, we have a
naturalinjectivemap K (C,K (p+q)) ⊂ K (X,K ).
n,1 C n,1 X
ProofofLemma1. Firstly,weremarkthatthereisanaturalinclusionofspacesofsections
H0(C,K (p + q)) ⊂ H0(X,K ). Indeed, the two spaces are both contained in the
C X
spaceofmeromorphicdifferentialsonC. ThusH0(X,K )identifiestothemeromorphic
X
differentials on C having poles of multiplicity one over the nodes, and regular outside
these points, and whose sums of residues over the nodes vanish. The inclusion above is
then adirect consequenceoftheResidueTheorem.
ThisinclusionyieldsthefollowinginjectionbetweentheKoszulcomplexesofK (p+
C
q) and K
X
0 → n+1H0(C,K (p+q)) → nH0(C,K (p+q))⊗H0(C,K (p+q)) → ...
V C V C C
↓ ↓
0 → n+1H0(X,K ) → nH0(X,K )⊗H0(X,K ) → ...
V X V X X
To conclude that this induces an injection on the degree 1 cohomology groups, we
usetheexistenceoftheretraction-homotopy(uptoacoefficient of(n+1)!)givenbythe
wedgeproduct:
n n+1
^H0(C,KC(p+q))⊗H0(C,KC(p+q)) → ^ H0(C,KC(p+q)).
ProofofTheorem 3. Step1. ConstructionofC and L.
We make use of the same curves as those used in [10]. Let S be a K3 surface whose
Picard group is generated by an ampleline bundleL of self-intersection L2 = 4k, where
g = 2k, and by arational curve∆such thatL.∆ = 2. We chooseX an irreduciblenodal
curve in the linear system |L|, having exactly one node as singularity. The curve X has
arithmeticgenus2k+1. WedenotebyC thenormalizationofX,pandq thetwodistinct
pointsofC lyingoverthenodeofX, and L = K (p+q). ThusC has genus2k.
C
Step 2. Recall thefollowingresult from[10]:
Theorem 6 Withthenotationabove,we haveK (S,L) = 0.
k,1
Applying the hyperplane section theorem we conclude that K (X,K ) = 0. By
k,1 X
meansofLemma1,weobtainK (C,L) = 0. Sincek = h0(L)−(k+1),itfollowsthat
k,1
thecurveC is ofmaximalgonalityk +1, andthat thegonalityconjectureisvalidforC.
3. Finalremarks
Remark 1 Using Theorem 4 one can see that for any of the curves C considered in the
proofs of Theorem 2, and Theorem 3, the vanishing Kh0(L′)−d,1(C,L′) = 0 predicted by
thegonalityconjectureholdsforanylinebundleL′ ofdegreeatleast3g. Thesameistrue
forgenericd-gonal curvesofgenusg,where g and dsatisfyg/3 < d < [g/2]+2.
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Remark 2 Thecaseg > (d−1)(d−2)has already been treated in[1]. Therefore, from
the viewpointof the Green-Lazarfeld conjecture for generic curves of fixed genus g, and
gonality d, for d ≥ 6, there is a gap remaining for 3d − 3 < g ≤ (d − 1)(d − 2) that
has to be solved differently. The case of maximal gonality d = k + 2 in the odd genus
case g = 2k +1, which seems to be the most difficult, is also left over. Nevertheless,we
should point out that in all these excepted cases, gonality conjecture is almost true, that
is, any line bundle L of degree at least 3g − 2 on a generic d-gonal curve C of genus g
(for any choice of g and d) satisfies Kh0(L)−d+1,1(C,L) = 0. This follows directly from
themainresultsof[8], [9], [10], andfrom Theorem 3of[1].
Remark 3 AnalternativeproofofLemma1canbeobtainedinamorealgebraicway,by
h g
factoring the normalization morphism through C → Y → X, where g is the smoothing
of the node x. We analyse the three morphisms between the structure sheaves, and we
obtain an isomorphism H0(X,K ) ∼= H0(Y,g∗K ), and an inclusion H0(C,K (p +
X X C
q)) ⊂ H0(Y,g∗K ). Denoting by W = H0(C,K (p+q)) insideH0(X,K ), we have
X C X
K (C,K (p+q)) ⊂ K (X,K ,W). Next,weusetheembeddingK (X,K ,W) ⊂
n,1 C n,1 X n,1 X
K (X,K ), which arisesfrom thespectralsequenceof[6], to conclude.
n,1 X
This shows that in Lemma 1 the fact of working over the complex numbers is not
essential,and thestatementis truefornodalcurvesoverany algebraicallyclosedfield.
Remark 4 The use of the results of [9], [10] in the proofs of our theorems shades a new
light on the relationships between Green’s conjecture, and the gonality conjecture, the
two statements seemed to be (in a somewhat misterious way) intimately related to each
other(seealso [1]). In fact, Lemma1 answers partiallyto theconjecturemadein[1].
References
[1] M. Aprodu, On the vanishing of higher syzygies of curves, Math. Z. 241 (2002)
1-15.
[2] M. Boratyn´sky and S. Greco, Hilbert functions and Betti numbers in a flat family,
Ann.Mat. PuraAppl.(4)142 (1985)277-292.
[3] S. Ehbauer, Syzygies of points in projective space and applications, Orecchia, Fer-
ruccio (ed.) et al., Zero-dimensionalschemes. Proceedings of theinternationalcon-
ference heldin Ravello,Italy,June8-13,1992.Berlin: deGruyter.(1994)145-170.
[4] W. Fulton, Hurwitz schemes and irreducibility of moduli of algebraic curves, Ann.
ofMath.90 (1969)541-575.
[5] M. Green, Koszul cohomology and the geometry of projective varieties, J. Diff.
Geom.19 (1984)125-171(withan Appendixby M.Green and R. Lazarsfeld).
[6] M. Green, Koszul cohomology and the geometry of projective varieties. II, J. Diff.
Geom.20 (1984)279-289.
[7] M. Green and R. Lazarsfeld, On the projective normality of complete linear series
on analgebraiccurve, Invent.Math.83 (1986)73-90.
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[8] M. Teixidor i Bigas, Green’s conjecture for the generic r-gonal curve of genus g ≥
3r −7, DukeMath. J.111 (2002)363-404.
[9] C. Voisin, Green’s generic syzygy conjecture for curves of even genus lying on a
K3 surface, J. EuropeanMath. Soc. 4 (2002)363-404.
[10] C. Voisin, Green’s generic syzygy conjecture for curves of odd genus, Preprint jan-
uary 2003.
ADRESSES
Marian Aprodu
Romanian Academy, Institute of Mathematics ”Simion Stoilow”, P.O.Box 1-764, RO-
70700,Bucharest, Romania(e-mail: [email protected])
and Universite´ de Grenoble 1, Laboratoire de Mathe´matiques, Institut Fourier BP 74, F-
38402Saint Martind’He`res Cedex, France (e-mail: [email protected])
Claire Voisin
Universite´ Paris7DenisDiderot-CNRSUMR7586-InstitutdeMathe´matiques2,Place
Jussieu,F-75251Paris Cedex 05 (e-mail: [email protected])
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