On Green and Green-Lazarfeld conjectures for simple coverings of algebraic curves
aa r X i v : . [ m a t h . AG ] S e p ON GREEN AND GREEN-LAZARFELD CONJECTURESFOR SIMPLE COVERINGS OF ALGEBRAIC CURVES
E. BALLICO AND C. FONTANARI
Abstract.
Let X be a smooth genus g curve equipped with a simple mor-phism f : X → C , where C is either the projective line or more generally anysmooth curve whose gonality is computed by finitely many pencils. Here weapply a method developed by Aprodu to prove that if g is big enough then X satisfies both Green and Green-Lazarsfeld conjectures. We also partiallyaddress the case in which the gonality of C is computed by infinitely manypencils. Introduction
Let X be a smooth complex curve of genus g . For any spanned L ∈ Pic( X ) and allintegers i, j let K i,j ( X, L ) denote the Koszul cohomology groups introduced in [10].Green’s conjecture states that K p, ( X, ω X ) = 0 if and only if p ≥ g − Cliff( X ) − X ) is the Clifford index of X , while Green-Lazarsfeld conjecture (see[11], Conjecture (3.7)) predicts that for every line bundle L on X of sufficientlylarge degree K p, ( X, L ) = 0 if and only if p ≥ r − gon( X ) + 1, where r is the(projective) dimension of L and gon( X ) is the gonality of X .Both Green and Green-Lazarsfeld conjectures have been verified for the generalcurve of genus g (see [16], [17], [7], [3]) and for the general d -gonal curve of genus g (see [15] for d ≤ g/
3, [16], Corollary 1 on p. 365, for d ≥ g/
3, [7], [4]). In particular,[4] shows that both conjectures are satisfied for any smooth d -gonal curve verifyinga suitable linear growth condition on the dimension of Brill-Noether varieties ofpencils. Such a condition holds for the general d -gonal curve, but for special curvesit turns out to be rather delicate (see [14], Statement T, and [6], Proposition 1.3).Here we consider the case in which X is a multiple covering. Let h : A → B be a covering of degree ≥ h is said to be simple if it does not factor non-trivially, i.e. for any smoothcurve D such that there are morhisms h : A → D and h : D → B with h = h ◦ h the morphism h is an isomorphism. Every covering of prime order is simple. Byapplying [4], Theorem 2, and [12], Theorem 1, we are going to prove the followingresult. Theorem 1.
Let X be a smooth genus g curve equipped with a simple morphism f : X → C of degree m ≥ , where C is a smooth curve of genus q whose gonality z is computed by finitely many g z . Assume g ≥ max { mq + ( m − mz − , mq + ( m − mz − , mz − , mz − } . Then X satisfies both Green andGreen-Lazarsfeld conjectures. Mathematics Subject Classification.
Key words and phrases.
Green conjecture; Green-Lazarsfeld conjecture; syzygy; covering.The authors are partially supported by MIUR and GNSAGA of INdAM (Italy).
In the special case q = 0, the corresponding notion of simple linear series isclassical (see for instance [1]) and Green’s conjecture has already been establishedfor m ≥ q = 0 our previous statement simplifies as follows. Corollary 1.
Let X be a smooth genus g curve carrying a simple g m of degree m ≥ . If g ≥ m − m − ≥ then X satisfies both Green and Green-Lazarsfeld conjectures. If instead q > C is computedby finitely many pencils, we obtain with the same method the following partialresult. Proposition 1.
Let X be a smooth genus g curve equipped with a simple morphism f : X → C of degree m ≥ , where C is a smooth curve of genus q ≥ and gonality z ≥ . Assume g ≥ mq + ( m − mz − . If m = 2 assume also g ≥ z − .Then K p, ( X, ω X ) = 0 for any p ≥ g − mz + 2 and K r − mz +2 , ( C, L ) = 0 for everyline bundle L on C with h ( C, L ) = r + 1 and deg( L ) ≥ g . Finally, if f : X → C is not simple, then f = f s ◦ · · · ◦ f with s ≥ f i a simple covering. One could hope to apply Theorem 1 to each covering f i , but thenumerical restrictions on the intermediate curves make such an iterative approacheffective only in very few cases.The authors are grateful to Marian Aprodu and Claire Voisin for stimulatinge-mail correspondence about Green and Green-Lazarsfeld conjectures.2. The proofs
Remark 1.
Let u : X ′ → C ′ be a degree m morphism between smooth curveswith X ′ of genus g and C ′ of genus q . Let v : X ′ → P be a degree x morphismsuch that the associated morphism ( u, v ) : X ′ → C ′ × P is birational onto itsimage. Then g ≤ mq + ( m − x −
1) (Castelnuovo-Severi inequality, see forinstance [13], Corollary at p. 26). Notice that ( u, v ) is not birational onto itsimage if and only if there are a smooth curve C ′′ (namely, the normalization of( u, v )( X ′ )) and morphisms w : X ′ → C ′′ , u : C ′′ → C ′ and v : C ′′ → P suchthat deg( w ) ≥ u = u ◦ w and v = v ◦ w . If u is simple, then u must bean isomorphism and ( u, v ) is not birational onto its image if and only if there isa morphism η = v ◦ u − : C ′ → P such that v = η ◦ u . Hence in the set-up ofTheorem 1 if g ≥ mq + ( m − mz −
1) then X has gonality mz and for every L ∈ Pic mz ( X ) such that h ( X, L ) = 2 there is R ∈ Pic z ( C ) such that h ( C, R ) = 2and L ∼ = f ∗ ( R ). Proof of Theorem 1.
By Remark 1, X has gonality mz and dim( W mz ( X )) =0. M. Aprodu proved that X has Clifford index mz − W mz + t ( X )) ≤ t for every integer t such that0 ≤ t ≤ g − mz +2 ([4], Theorem 2). Since the function x → dim( W x ( X )) is strictlyincreasing in the interval [gon( X ) , g − W g − mz +2 ( X ))= g − mz +2. Assume dim( W g − mz +2 ( X )) > g − mz +2, i.e. dim( W g − mz +2 ( X )) = g − mz − j for some integer j ≤ mz −
3. We have j ≥ g ≥ j + 3 and 2 j + 2 ≤ g − mz + 2 ≤ g − − j , hence a theorem of R. Horiuchi yields dim( W j +2 ( X )) = j ([12], Theorem 1). Let now Γ be any irreducible component of W j +2 ( X ) such REEN AND GREEN-LAZARFELD CONJECTURES FOR COVERINGS 3 that dim(Γ) = j . Since f is simple and g − mq > ( m − mz − ≥ ( m − j + 1), by Remark 1 there are an integer y ≤ ⌊ (2 j + 2) /m ⌋ , a non-emptyopen subset Φ of Γ and an open subset Ψ of W y ( C ) such that every elementof Φ is the pull-back of an element of Ψ plus 2 j + 2 − my base points. Thus j = dim(Γ) = dim(Ψ) + 2 j + 2 − my ≤ dim( W y ( C )) + 2 j + 2 − my . We have Φ = ∅ ,so Ψ = ∅ and y ≥ z . Hence dim( W y ( C )) ≤ dim( W z ( C )) + 2( y − z ) ([9], Theorem1). Since by assumption dim( W z ( C )) = 0, by putting everything together we get j ≤ y − z ) + 2 j + 2 − my ≤ j + 2 − mz , i.e. j ≥ mz −
2, contradiction. (cid:3)
The following auxiliary result provides a suitable generalization of [4], Theorem2, by repeating almost verbatim the same proof.
Lemma 1.
Fix an integer n ≥ and let C be a smooth d -gonal curve of genus g such that dim G d + m ≤ n − m for all m with n − ≤ m ≤ g − d + n + 1 . Then K g − d + n, ( C, ω C ) = 0 and K r − d + n, ( C, L ) = 0 for every line bundle L on C with h ( C, L ) = r + 1 and deg( L ) ≥ g .Proof. Define integers k, ν as follows: k = g − d + n (1) ν = 2 k − g (2)and let X be the stable curve obtained from C by identifying ν + 1 pairs of generalpoints on C . In particular, let p, q be a pair of points on C identified to a nodeon X . If K k, ( C, ω C ( p + q )) = 0 then according to [7], Theorem 2.1, for everyeffective divisor E of degree e ≥ K k + e, ( C, ω C ( p + q + E )) = 0. Thusif L is any line bundle on C of degree x ≥ g , then h ( C, L − ω C ( p + q )) ≥ K k + x − g, ( C, L ) = 0. On the other hand, by [7], Lemma 2.3 and [16], p. 367, wehave K k, ( C, ω C ) ⊆ K k, ( C, ω C ( p + q )) ⊆ K k, ( X, ω X ), therefore in order to proveour statement we may assume K k, ( X, ω X ) = 0 and look for a contradiction. By (2), X has genus 2 k +1, hence by [3], Proposition 8, there exists a torsion-free sheaf F on X with deg( F ) = k + 1 and h ( X, F ) ≥
2. Let s with 0 ≤ s ≤ ν + 1 be the numberof nodes at which F is not locally free. If f : X ′ → X is the partial normalizationof X at all such nodes, then F = f ∗ ( L ), where L = f ∗ ( F ) / Tors( f ∗ ( F )) is a linebundle on X ′ with deg L = k + 1 − s and h ( X ′ , L ) = h ( X, F ) ≥
2. By taking thepull-back of L on C , we obtain a g k +1 − s not separating ν + 1 − s pairs of generalpoints on C , hence it follows that dim G k +1 − s ( C ) ≥ ν + 1 − s .In order to reach a contradiction, assume first 0 ≤ s ≤ g − d + 2 (noticethat if g = 2 r − d = r + 1 this case does not occur). From (1) we obtain k + 1 − s = d − d + g + n + 1 − s with n − ≤ − d + g + n + 1 − s ≤ g − d + n + 1.Hence our numerical hypotheses imply thatdim G k +1 − s ( C ) ≤ g − d + 2 n − s ≤ ν − s. Assume now s > g − d + 2. We claim that also in this casedim G k +1 − s ( C ) = max r { r −
1) + dim W rk +1 − s ( C ) } < ν + 1 − s. Indeed, we havedim W rk +1 − s ( C ) ≤ dim W k +1 − s − ( r − ( C ) ≤≤ dim W d ( C ) + 2( k + 1 − s − ( r − − d ) ≤≤ k + 1 − s − ( r − − d ) E. BALLICO AND C. FONTANARI where the second inequality is provided by [9], Theorem 1. Hence from (2) it followsthat dim W rk +1 − s ( C ) < ν + 1 − s − r −
1) for any r , as claimed. (cid:3) Proof of Proposition 1.
We argue as in the proof of Theorem 1 by applyingLemma 1 with n = 2 instead of [4], Theorem 2. This time we need to provedim( G g − mz +3 ( X )) ≤ g − mz +4, so we assume by contradiction dim( G g − mz +3 ( X ))= g − mz + 1 − j with j ≤ mz −
4. Once again the numerical hypotheses of [12],Theorem 1, are easily checked, hence we get j ≤ dim( W z ( C )) + 2 j + 2 − mz . Sincein any case dim( W z ( C )) ≤ j ≥ mz − (cid:3) References [1] R. D. M. Accola: On Castelnuovo’s inequality for algebraic curves. I. Trans. Amer. Math.Soc. 251 (1979), 357–373.[2] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris: Geometry of algebraic curves. Vol.I. Grundlehren der Mathematischen Wissenschaften, 267. Springer-Verlag, New York, 1985.[3] M. Aprodu: Green-Lazarsfeld gonality conjecture for a generic curve of odd genus. Int. Math.Res. Not. 2004, no. 63, 3409–3416.[4] M. Aprodu: Remarks on syzygies of d -gonal curves. Math. Res. Lett. 2 (2005), 387–400.[5] M. Aprodu and G. Farkas: Koszul cohomology and applications to moduli, in: Aspects ofvector bundles and moduli, Clay Mathematical Institute, Clay Math. Proc. Vol. 10, AmericanMath. Soc. (2009). Pre-print arXiv:0811.3117.[6] M. Aprodu and G. Pacienza, The Green conjecture for exceptional curves on a K r -gonal curve of genus g ≥ r − K Department of Mathematics, University of Trento, Via Sommarive 14, 38123 Povo(TN), Italy
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