Exceptional curves on Del Pezzo surfaces
Andreas Leopold Knutsen∗1
1 Universit¨at Essen, Fachbereich 6 – Mathematik, 45117 Essen, Germany
Received 3 September 2001, revised 27 June 2002, accepted 2 August 2002 Published online 25 June 2003
Key words Line bundles, curves, Del Pezzo surfaces, exceptional curves, gonality, Clifford index MSC (2000) Primary: 14J26; Secondary: 14H51.
We classify all cases of exceptional curves on Del Pezzo surfaces, which turn out to be the smooth plane curves and some other cases with Clifford dimension3. Moreover, the property of being exceptional holds for all curves in the complete linear system. We use this study to extend the results of Pareschi [17] on the constancy of the gonality and Clifford index of smooth curves in a complete linear system on Del Pezzo surfaces of degrees≥ 2to the case of Del Pezzo surfaces of degree1, where we explicitly classify the cases where the gonality and Clifford index are not constant.
1 Introduction
The two main purposes of this paper are (1) to classify all cases of exceptional curves on Del Pezzo surfaces and (2) to study the gonality and Clifford index of linearly equivalent smooth curves in|L|on a Del Pezzo surface of degree1.
In the past decades, several authors have studied the question whether exceptional linear series on a curveC on a certain surface propagate to the other members of|C|. For instance, for aK3surface, Saint–Donat proved in [20] thatCpossesses ag12or ag31if and only if every smooth curve in|C|does, and Reid [19] extended this result to othergd1s. Harris and Mumford conjectured that linearly equivalent smooth curves on aK3surface have the same gonality, but a counterexample of Donagi and Morrison [6] proved this to be false. This led Green [10] to modify the conjecture to the effect that linearly equivalent smooth curves on aK3surface have the same Clifford index. This was proved by Green and Lazarsfeld in the famous paper [9]. Moreover, the Donagi–Morrison counterexample is still the only example known where the gonality is not constant, and Ciliberto and Pareschi [2]
proved that this is indeed the only counterexample ifOS(C)is ample.
The results of Green and Lazarsfeld were extended to Del Pezzo surfaces of degree≥2by Pareschi in [17].
In fact, he showed that fordegS≥2the Clifford index of the smooth curves in a linear system|L|containing a smooth curve of genusg≥4(which is equivalent toLbeing nef ) is constant, and that the gonality also is when g≥2, except for the following case [17, Example (2.1)]:
Case (I):degS= 2andL∼ −2KS(in particularg(L) = 3). In this case there is a codimension1family of smooth hyperelliptic curves in|L|, whereas the general smooth curve is trigonal.
More precisely, denote byφ:S→P2the morphism defined by−KS, which is a double cover ramified along a smooth quartic. Then
H0(L) = φ∗H0
OP2(2)
⊕W ,
whereW is the1–dimensional subspace of sections vanishing on the ramification locus. The smooth curves in the first summand are double covers of conics, whence hyperelliptic, whereas the general curve in the linear system maps isomorphically to a smooth plane quartic and is therefore trigonal.
∗ e–mail:[email protected]
c 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 0025-584X/03/25607-0058 $ 17.50+.50/0
By work of Serrano [21] there is an example whendegS= 1 (namelyL∼ −nKS, forn≥3)where neither the gonality nor the Clifford index is constant (see Example 3.8 below). This shows that the situation is more subtle fordegS= 1.
In this paper we first study exceptional curves on Del Pezzo surfaces of any degree. These are the curves whose Clifford index is not computed by a pencil, and are conjectured to be extremely rare. In fact Eisenbud, Martens, Lange and Schreyer [8] conjecture that the only types of exceptional curves are smooth plane curves, and certain curves of degree4r−3and genus4r−2inPrforr≥3. We show (Theorem 1.1 (c) below) that exceptional curves on a Del Pezzo surface are exactly the strict transforms of smooth plane curves (whence of Clifford dimension2), and of smooth curves in| −3KS|fordegS= 3(of Clifford dimension3). This is in full accordance with the conjecture just mentioned. Furthermore, we show that ifCis an exceptional curve andAa line bundle onCcomputing its Clifford dimension, thenA=OC(D)for a line bundleDon the surface and any smooth curveC ∈ |C|is exceptional with its Clifford dimension computed byOC(D).
As an application of this study, we classify all the cases fordegS = 1where the gonality and the Clifford index are not constant, which will include the example of Serrano.
More precisely, we will show that the Clifford index is constant whendegS= 1, except precisely for the case (III) below, and that the gonality is constant except precisely for the cases (II) and (III) below:
Case (II):degS= 1andL∼ −2KS+ 2Γ, for a(−1)–curveΓ (in particularg(L) = 3). In this case there is also a codimension1family of smooth hyperelliptic curves in|L|, whereas the general smooth curve is trigonal.
Case (III):degS = 1,Lis ample,L .Γ≥2for all(−1)–curvesΓifL2 ≥8, and there is an integerd≥3 such thatL2≥5d−8≥7,L . KS =−dandL . C≥dfor all smooth rational curvesCwithC2= 0.
In this case there is a codimension1 family of|L| of smooth(d−1)–gonal curves(which are exactly the smooth curves passing through the base point ofKS), whereas the general smooth curve isd–gonal.
Ifg(L) ≥ 4, then the smooth curves in the codimension1 family have Clifford indexd−3, whereas the general smooth curve has Clifford indexd−2.
The three easiest examples of Case (III) for high genus areL∼ −dKSford≥3(precisely the case described by Serrano [21], see Example 3.8 below),L ∼ −(d−1)KS + Γ, ford ≥ 3 and a(−1)–curveΓ, andL ∼
−(d−2)KS+R, ford≥4and a smooth rational curveRwithR2= 0. In the case of lowest genus,g(L) = 3 (andL2= 7)andL∼ −2KS+ Γfor a(−1)–curveΓ.
Since the Picard group of a Del Pezzo surface is well–known, the conditions in Case (III) pose restrictions on the coefficients of the generators ofPicSin the representation ofL. We thus obtain a precise description of the line bundles in Case (III). This is given at the end of Section 4.
The main results obtained in this paper are summarized in Theorem 1.1 below. For the notation, ifLis a nef line bundle on a Del Pezzo surface, we define the (finite) set of curvesR(L)by:
R(L) := {Γ| Γ = (−1)–curve and Γ. L= 0},
and denote its cardinality bym(L). (Note that by the Hodge index theorem, the curves inR(L)are necessarily disjoint.) Moreover, see Section 2.3 for the definition ofSn.
Theorem 1.1 LetLbe a nef line bundle on a Del Pezzo surfaceSwithg(L)≥2.
(a)All the smooth curves in|L|have the same gonality if and only ifLis not as in the cases(I), (II)or(III) above.
(b)Ifg(L)≥4, then all the smooth curves in|L|have the same Clifford index if and only ifLis not as in case (III)above.
(c)Ifg(L)≥4, then either all or none of the smooth curves in|L|are exceptional, and if they are, they all have the same Clifford index and Clifford dimension. Furthermore, the curves are exceptional of Clifford dimension r≥2and Clifford indexcif and only ifLis as in one of the two following cases:
(i)There exists a base point free line bundleDonSsatisfying
D2 = 1, D.KS = −3, D . L = c+ 4 and Γ. L ≤ 1 for any Γ∈ R(D) (the latter condition is equivalent toDCbeing very ample for allC∈ |L|).
In this caser = 2, the Clifford dimension is computed only byOC(D)for all smooth curvesC ∈ |L|and φD :S→P2gives the identificationSSm(D)and maps every smooth curve in|L|to a smooth plane curve of degreeD . L=c+ 4, whence all these curves are exceptional of Clifford dimension2and Clifford indexc.
(ii)SSnforn∈ {6,7,8}and L ∼ −3KSn+
n−6
i=1
aiΓi, Γi = (−1)–curve, ai∈ {2,3}.
In this caser= 3,g(L) = 10,c= 3and for all smooth curvesC ∈ |L|, the Clifford dimension is computed only byOC(D), forD:=−KSn+n−6
i=1 Γi.
We actually prove a more precise result. In fact, any line bundleAcomputing the gonality of a smooth curve on a Del Pezzo surface, is of a particular form (it is, so to speak, induced from a line bundle on the surface), and we are able to find all possible such in Proposition 5.1 below.
We note that the overall pattern here is that the lower the degree ofSis
equivalently, the more points ofP2 we blow up
, then the more “special” examples we get. In fact, up todegS = 4, all linearly equivalent smooth curves have the same gonality and Clifford index and the only exceptional curves are the smooth plane curves. For degS = 3, new examples of exceptional curves turn up, fordegS = 2, the first example of linearly equivalent smooth curves of different gonalities appears, and fordegS = 1, more such examples turn up, together with examples of linearly equivalent smooth curves of different Clifford indices.
It would be an interesting problem, to study the picture when one blows up≥ 9 points ofP2 (in general position), to see which new examples might turn up. Unfortunately, the methods in this paper rely upon−KS
being ample (or at least effective) and at the moment we do not see how to treat higher blow ups ofP2. The paper is organised as follows.
First, in Section 2 we gather basic definitions and results that will be needed throughout the paper.
We classify all exceptional curves on Del Pezzo surfaces in Section 3, thus finishing the proof of Theorem 1.1 (c). The methods are similar to the ones used by Pareschi [17], but due to the extension to Del Pezzo surfaces of degree1, we need a more careful analysis. We also use results about exceptional curves in [8].
As an application we prove parts (a) and (b) of Theorem 1.1 in Section 4. The result relies upon the partial result obtained in [11, Section 4].
Finally, in Section 5 we give the complete list of all the possible pencils computing the gonality of a smooth curve on a Del Pezzo surface.
Note that parts (a) and (b) of Theorem 1.1 fordegS ≥ 2 are equal to the results of Pareschi [17]. In our proofs, we however never assume thatdegS = 1, both to give a more complete exposition, and also because restricting to the casedegS = 1would not make the proofs easier, since the casedegS = 1is exactly the case where most problems show up.
Remark 1.2 The smooth curves in case (II) above are simply the strict transforms of smooth curves in|−2KS| fordegS= 2which do not pass through the blown up point.
The strict transforms of such curves passing through the blown up point are the curves in case (III) with L∼ −2KS+ Γ.
2 Background material
2.1 Basic notation and definitions
The ground field is the field of complex numbers. All surfaces aresmooth irreducible algebraic surfaces.
ByBP1,...,Pn(S)we will mean the blowing up ofSat the pointsP1, . . . , Pn.
By acurveon a surfaceSis always meant areduced and irreducible curve(possibly singular), i.e. a prime divisor. Line bundles and divisors are used with little or no distinction, as well as the multiplicative and additive notation. Linear equivalence of divisors is denoted by∼.
IfLis any line bundle on a surface,Lis said to benumerically effective, or simplynef, ifL . C ≥0for all curvesConS. In this caseLis said to bebigifL2>0. Moreover we writeL≥0forLeffective andL >0for Leffective and nonzero.
IfFis any coherent sheaf on a varietyV, we shall denote byhi(F)the complex dimension ofHi(V,F), and byχ(F)the Euler characteristic
(−1)ihi(F).
Thesectional genusof a line bundleLis the integerg(L) =12L .(L+KS) + 1. IfD∈ |L|, theng(L)is the arithmetic genus ofD.
IfCis a curve on a surface andLis a line bundle on the surface, we often use the notationLC:=L⊗ OC. 2.2 k–very ampleness
A line bundleLon a smooth varietyX is said to bek–very ample (resp.k–spanned), for an integerk ≥ 0, if for each (resp. each curvilinear)0–dimensional subschemeZofX of lengthh0
OZ
=k+ 1, the restriction map H0(L) → H0
L ⊗ OZ
is surjective. (Recall that a 0–dimensional scheme Z is called curvilinear if dimTxZ ≤1for everyx∈ Zred.)
In [12] we made the following definition: A line bundleLon a smooth surfaceSisbirationallyk–very ample (resp.birationallyk–spanned), for an integerk ≥ 1, if there exists a non–empty Zariski–open subsetU ofS such that the restriction mapH0(L)→H0
L⊗ OZ
is surjective for any (resp. any curvilinear)0–dimensional subschemeZofSof lengthh0
OZ
=k+ 1with support inU. 2.3 Del Pezzo surfaces
A surfaceS is called a Del Pezzo surface if its anticanonical bundle−KSis ample. The degree ofSis defined asdegS=KS2. We have the following classification of such surfaces:
Proposition 2.1 [4]LetSbe a Del Pezzo surface. Then (a) 1≤degS≤9,
(b) degS= 9if and only if SP2,
(c) if degS = 8, then eitherSP1×P1orS BP
P2 , (d) if 1≤degS≤7, thenSBP1,...,P9−degS
P2 .
We will often denote a Del Pezzo surface of degreedegSbySn, wheren= 9−degS. Letπ:S →P2be the blowing up as in (d). Denote bylthe class ofπ∗
OP2(1)
and byeithe class ofπ−1(Pi). We then have l2 = 1, ei. ej = −δij, ei. l = 0,
and
PicSn Zl⊕Ze1⊕. . .⊕Zen Zn+1. Therefore any line bundle onSnis of the formal−n
1bieiand−KS= 3l−n 1ei.
A(−1)–curve on a Del Pezzo surface is a curveΓsatisfyingΓ2 =−1. Such a curve is necessarily smooth and rational and satisfiesΓ. KS =−1(by the adjunction formula), and those are the only curves with negative self–intersection on a Del Pezzo surface. It follows that a line bundleD≥0is not nef if and only if there exists a(−1)–curveΓsuch thatΓ. D <0.
The number of(−1)–curves on a Del Pezzo surface is finite by standard arguments.
Note that a(−1)–curve is frequently also calledexceptional, since it is the exceptional divisor of a suitable blowing up. We will however not use this expression for(−1)–curves, since the termexceptional curveswill be reserved for another particular type of curves (see Section 2.4 below).
We gather some basic properties of line bundles on a Del Pezzo surface:
(P1) | −KS|is base point free unlessS=S8, in which case it has exactly one base point, namely the ninth point of intersection of two independent plane cubics based on9points.
(P2) | −KS|is very ample if and only ifdegS ≥3.
(P3) IfCis a curve, thenh1(−C) =h1(C+KS) = 0[17, (0.4.5)].
(P4) Lis nef if and only if it is base point free orL∼ −KS8 [5, (Cor. 4.7)].
(P5) IfLis base point free, thenh1(L) =h2(L) = 0[17, (0.4.6)].
(P6) IfL > 0, L ∼ −KS8 andLis not a(−1)–curve onS, then−L . KS ≥ 2 (Hodge index theorem and adjunction formula).
(P7) IfLis nef withL2>0we havehi(L) =hi(L+KS) = 0, fori= 1,2,h0(L+KS) =g(L)≥1and the general member of|L|is smooth and irreducible.
To show the last condition, note thath1(L) =h2(L) = 0by (P5) andh2(L+KS) =h0(−L) = 0by Serre duality sinceL >0. Moreover,Lis numerically1–connected by the Hodge index theorem
sinceL2 >0 and soh1(L+KS) = 0by Ramanujam vanishing. By (P1) and (P4)Lis either base point free or has only one base point, so smoothness and irreducibility follow from Bertini’s theorem.
2.4 Clifford index, gonality and the vector bundlesE(C, A)
LetCbe a smooth irreducible curve of genusg≥2. We denote bygdra linear system of dimensionrand degree d. The gonality gonCofCis defined as
gonC := min
k |C has ag1k .
In particular,Cis calledhyperellipticif gonC = 2andtrigonalif gonC = 3. Note that if gonC =k, allg1ks must necessarily be base point free and complete.
Ifg≥4andAis a line bundle onC, then theClifford indexofAis the integer CliffA := degA−2
h0(A)−1 and theClifford index ofCitself is defined as
CliffC := min
CliffA |h0(A)≥2, h1(A)≥2 .
Clifford’s theorem then states thatCliffC ≥0with equality if and only ifCis hyperelliptic andCliffC = 1if and only ifCis trigonal or a smooth plane quintic.
At the other extreme, we get from Brill–Noether theory (cf. [1, V]) that the gonality ofCsatisfies gonC ≤ g+32 , whenceCliffC≤ g−12 . For the general curve of genusg, we haveCliffC=g−12 .
We say that a line bundleAonCcontributes to the Clifford index ofCif bothh0(A)≥2andh1(A) ≥2, and that itcomputes the Clifford index ofCif in additionCliffC= CliffA.
Note thatCliffA= Cliff
ωC⊗A−1
and also observe that by Riemann–Roch
h0(A) +h1(A) = g+ 1−CliffA . (2.1)
TheClifford dimensionofCis defined as min
h0(A)−1 | Acomputes the Clifford index ofC .
A line bundleAwhich achieves the minimum and computes the Clifford index, is said tocomputethe Clifford dimension. A curve of Clifford indexcis(c+ 2)–gonal if and only if it has Clifford dimension1. For a general curveC, we have gonC=c+ 2.
Lemma 2.2 [3, Theorem 2.3]The gonalitykof a smooth(irreducible)projective curveCsatisfies CliffC+ 2 ≤ k ≤ CliffC+ 3.
The curves satisfying gonC = CliffC+ 3 are conjectured to be very rare and calledexceptional(cf. [8]
or Proposition 3.11 and Remark 3.12 below). The curves of Clifford dimension2are exactly the smooth plane curves of degree≥5.
LetCbe a smooth curve on a regular surfaceS
i.e.h1(OS) = 0
.Recall that if Ais a line bundle onC which is generated by its global sections, then one can associate to the pair(C, A)a vector bundleE(C, A)of rankh0(A)as follows (this vector bundle was introduced by Lazarsfeld in [13], for more details we refer to [2]
and [17]). Thinking ofAas a coherent sheaf onS, we get a short exact sequence
0 −→ F(C, A) −→ H0(A)⊗COS −→ A −→ 0 (2.2)
ofOS–modules, whereF(C, A)is locally free(sinceAis locally isomorphic toOCand hence has homological dimension one overOS).
Dualizing this sequence, one gets
0 −→ H0(A)∨⊗ OS −→ E(C, A) −→ NC/S⊗A∨ −→ 0, (2.3) whereE(C, A) :=F(C, A)∨. This vector bundle has the following properties:
detE(C, A) = OS(C), (2.4)
c2(E(C, A)) = degA , (2.5)
rkE(C, A) = h0(A), (2.6)
hi(E(C, A)∨) = h2−i
E(C, A)⊗ωS
= 0, i = 0,1, (2.7)
h0(E(C, A)) = h0(A) +h0
NC/S⊗A∨
, (2.8)
IfA≤ NC/S, thenE(C, A)is generated by its global sections off a finite (2.9) set coinciding with the (possibly zero) base divisor ofNC/S⊗A∨.
Note that ifAis any line bundle onCcomputing the gonality or the Clifford index ofC, thenAis generated by its global sections, and we can carry out the construction of the vector bundleE(C, A)above.
Specializing to Del Pezzo surfaces, we get from (2.3) tensored byωSthat h1(A) = h0
E(C, A)⊗ωS
. (2.10)
Moreover, we haveNC/S⊗A∨ωC⊗A∨⊗ OC(−KS), and by (P1) we get:
Lemma 2.3 AssumeAandωC⊗A∨are both base point free(as will be the case ifAcomputes the Clifford index of C).
ThenE(C, A)is generated by its global sections except possibly at the base pointxof | −KS|, whenKS2= 1 andCpasses throughx.
3 Exceptional curves
We will in this section study the exceptional curves on a Del Pezzo surface. Clearly, on a Del Pezzo surface of typeSn, all the strict transforms of the smooth plane curves are exceptional. The following is a non–trivial example of smooth curves of Clifford dimension3.
Example 3.1 LetSS6and consider the line bundleL∼ −3KS6. All the smooth curves in|L|are of genus 10, and for any suchC∈ |L|, we haveh0
OC
−KS6
=h1 OC
−KS6
= 4, soOC
−KS6
contributes to the Clifford index ofC. We compute
CliffOC
−KS6
= degOC
−KS6
−2 h0
OC
−KS6
−1
= 3.
Now assumeCis not exceptional of Clifford index3. Thend:=gonC≤5, and by [11, Prop. 3.7] there has to exist an effective divisorDsatisfyingD . L−D2=dand such thatDis of one of the following types:
(a) D2= 0, −D . KS6 = 2, (b) D2= 1, −D . KS6 = 3, (c) D∼ −KS6.
In these three cases we get respectivelyD . L−D2=−3D . KS6−D2= 6,8,6, whence a contradiction.
Hence all the smooth curves in|L|have gonality6and Clifford index3, which shows that they are exceptional.
If the Clifford dimension of any suchCis2, thenCis isomorphic to a smooth plane septic, and we easily get a contradiction from the genus formula of a smooth plane curve. So all the curves have Clifford dimension3.
We show in this section that the only exceptional curves on a Del Pezzo surface are precisely the strict trans- forms of the smooth plane curves and of the curves in the above example.
First we need the following result, which we formulate in a general setting:
Proposition 3.2 LetRbe a vector bundle of rank≥2on a surfaceS generated by its global sections off a finite set and such thatH0(R∨) = 0andc1(R)2≥0.
(a)Ifc1(R)2>0, thenc2(R)−2rkR≥min{−2, c1(R). ωS}. If furthermoreh0(ωS) = h1(ωS) = 0and h0(R⊗ωS)>0, thenc2(R)−2rkR≥ −2.
(b) Ifc1(R)2= 0andh1(OS) = 0, thendetR∼ OS(kΣ)for a base point free pencil|Σ|onSand an integer k≥1, andc2(R)−2rkR≥kmin{−2,Σ. ωS}.
P r o o f. Let r := rkR. For a general r–dimensional subspace V ⊆ H0(R), the evaluation map eV : V ⊗ OS → Ris generically an isomorphism and has rank exactlyr−1along a divisorD ∈ |detR|defined by the vanishing ofdeteV. By Bertini’s theorem,Dis smooth except possibly at the points whereRfails to be generated. Moreover,B :=cokereV is a torsion free sheaf onD(which is a line bundle outside of the singular points ofD), and we have the exact sequence:
0 −→ V ⊗ OS −→R −→B −→0 (3.1)
ofOS–modules. Dualizing this sequence, one obtains
0 −→ R∨ −→ V∨⊗ OS −→ A −→ 0, (3.2)
whereA:=HomOD
B,OD(D)
is also torsion free. Moreover we find from (3.1) and (3.2) that
c2(R) = degA , (3.3)
rkR ≤ h0(A), (3.4)
degB = 2pa(D)−2−degA−D . KS, (3.5)
where the degree ofA is defined by the Riemann–Roch type formuladegA := χ(A)−χ(OD) = χ(A) + pa(D)−1, and similarly forB.
To prove (a), let nowD2 >0. Then by Bertini’s theorem,Dis reduced and irreducible (i.e. what we call a curve).
Ifh1(A)>0, then by Clifford’s theorem for torsion free rank1sheaves on singular curves (see the appendix of [7]),CliffA≥0, whence by (3.3) and (3.4)
c2(R)−2rkR ≥ degA−2h0(A) ≥ CliffA−2 ≥ −2,
which is what we want to prove. Also note thath1(A) =h0(R⊗ωS)whenh0(ωS) =h1(ωS) = 0.
Ifh1(A) = 0, thendegA =h0(A) +pa(D)−1. By (3.1)Bis generated by its global sections off a finite set, whenceh0(B)>0anddegB ≥0. Combining (3.3), (3.4) and (3.5) we get
c2(R)−2rkR ≥ degA−2h0(A) = −degA−2 + 2pa(D) = degB+D . KS ≥ D . KS, which finishes the proof of (a).
Finally we prove (b), so letD2 = 0. Since the evident mapΛrH0(R) → ΛrR =OS(D)is surjective off a finite set, |D| is a linear system without fixed components, and is therefore base point free, sinceD2 = 0.
Since we assumeS to be regular, it follows that|D|is composed with a rational pencil, i.e. there is a smooth curveΣ ⊆ S and an integerk ≥ 1such thatD ∼ kΣandD is the disjoint union ofkcurves in|Σ|, write D=D1+. . .+Dk.
The sheaves AandB are therefore the direct sumsA = ⊕Ai andB = ⊕Bi of sheaves on eachDi. By arguing as in the proof of (a) we find thatdegAi−2h0(Ai)≥min{−2, Di. KS}, whence
c2(R)−2rkR ≥ degA−2h0(A) = degAi−2h0(Ai)
≥ kmin{−2,Σ. KS}. This concludes the proof of (b).
In particular, by specializing to a Del Pezzo surface, using (P6) and the fact thath0(kΣ) = k+ 1by (P5), we get
Corollary 3.3 With the same assumptions as in Proposition3.2, assume thatSis a Del Pezzo surface. Set D:= detR. Then
(a) c2(R)−2rkR≥ −2, whenD∼ −KS8,
(b) c2(R)−2rkR≥ −2r, forr:=h0(D)−1, whenD2= 0, (c) c2(R)−2rkR≥D . KS= 2g(D)−D2−2otherwise.
Furthermore, in case(c), ifh0(R⊗ωS)>0, then we have the stronger inequalityc2(R)−2rkR≥ −2.
We now make the following assumptions:
(∗) LetC0be a smooth curve of genusg≥4on a Del Pezzo surface. SetL:=OS(C0). LetAbe a base point free line bundle onC0such that3≤h0(A)≤h1(A) (this impliesdegA≤g−1)andωC0⊗A∨is base point free.
As in Subsection 2.4 we get the vector bundleE :=E(C0, A), which is of rankh0(A)≥3. Since by (2.10) h0(E⊗ωS)>0, we can find an invertible subsheaf0→M →E, and after saturating, we can assume we have an exact sequence
0 −→ M −→ E −→ R −→ τ −→ 0, (3.6)
whereRis locally free of rank≥2andτis supported on a finite set.
DefineD:= detR. We have
Lemma 3.4 (a)h0(R∨) =h1(R∨) = 0,
(b)D is non trivial and eitherD ∼ −KS8 or D is base point free. In particularh1(D) = h2(D) = 0.
MoreoverL∼M +D.
P r o o f. Similar to the proof of [17, (1.12)].
For anyC∈ |L|we writeDC:=OC(D).
Proposition 3.5 (a) h0(DC)is the same for anyC∈ |L|, (b) DC0 ≥A,
(c) h1(DC) =h0(M⊗ωS)>0for anyC∈ |L|,
(d)CliffDCis the same for anyC∈ |L|andCliffDC≤CliffA.
P r o o f. The statements (a) – (c) follow as in the proof of [17, (1.13)]. To prove (d), note thatRis globally generated off a finite set, so Corollary 3.3 applies.
We have
CliffA ≥ D . M+c2(R)−2rkR . (3.7)
Indeed,
CliffA = c2(E)−2(rkE−1) (by (2.5) and (2.6))
= D . M+c2(R) +lengthτ−2rkR (by (3.6))
≥ D . M+c2(R)−2rkR . IfD2= 0, then by (3.7) and (b) of Corollary 3.3:
CliffA ≥ D . L−2r . (3.8)
At the same time, by the exact sequence
0 −→ −M −→ D −→ DC −→ 0 (3.9)
and Lemma 3.4,
h0(DC) = h0(D) +h1(−M) ≥ r+ 1. (3.10)
Combining (3.8) and (3.10) we get the desired resultCliffA≥CliffDC. Assume nowD2>0. We have
h0 DC
= h0(D) +h1(−M) ≥ h0(D) = 1
2D .(D−KS) + 1, (3.11) whence
CliffDC = D . L−2 h0
DC
−1
≤ D . M +D . KS. (3.12)
IfD∼ −KS8, thenh0(D) = 2, and sinceDC0 ≥Aandh0(A)≥3, we haveh0 DC
=h0 DC0
≥3, so we have strict inequalities in (3.11) and (3.12).
Now the result follows by combining (3.7), (3.12) and parts (a) and (c) of Corollary 3.3.
We need the following easy
Lemma 3.6 LetLbe a nef line bundle on a Del Pezzo surface withg(L)≥4. AssumeL+KS ∼D1+D2 for two divisorsD1andD2satisfyingh0(Di)≥2. ThenOC(D1)andOC(D2)contribute to the Clifford index of any smoothC∈ |L|and
CliffOC(D1) = CliffOC(D2) ≤ D1. D2,
with equality if and only ifh0(OC(Di)) =h0(Di)andh1(Di) = 0fori= 1,2.
Furthermore, if equality holds and there exists a smooth curveC0 ∈ |L|such thatCliffC0 =D1. D2, then any(−1)–curveΓwithΓ. L= 0will satisfy(Γ. D1,Γ. D2) = (0,−1)or(−1,0).
P r o o f. From the exact sequence
0 −→ KS−D2 −→ D1 −→ OC(D1) −→ 0,
h0(OC(D1))≥h0(D1)≥2andh1(OC(D1))≥h2(KS−D2) =h0(D2)≥2, so thatOC(D1)contributes to the Clifford index of any smoothC∈ |L|. By symmetry we get the same result forD2.
NowOC(D2) OC(L+KS−D1)ωC⊗ OC(D1)∨, soCliffOC(D1) = CliffOC(D2).
One computes by Riemann–Roch onS:
CliffOC(D1) = D1. L−2
h0(OC(D1))−1
≤ D1. L−2
h0(D1)−1
≤ D1. L−D1.(D1−KS) = D1.(L−D1+KS)
= D1. D2.
It is clear that equality holds if and only ifh0(OC(Di)) =h0(Di)andh1(Di) = 0fori= 1,2.
For the last statement, since−1 = Γ.(L+KS) = Γ. D1+ Γ. D2, assume by symmetry to get a contradiction thatΓ. D1>0. We then getΓ. D2=−1−Γ. D1≤ −2. Clearlyh0(D1+ Γ)≥h0(D1)≥2andh0(D2−Γ) = h0(D2)≥2, so by what we have just shownOC(D1+Γ)contributes to the Clifford index of any smoothC∈ |L|, and
CliffOC0(D1+ Γ) ≤ (D1+ Γ).(D2−Γ) = D1. D2−2Γ. D1 < CliffC0, a contradiction.
As in [11] we say that a line bundleDsatisfies the conditions(†)for an integerk≥1if:
h0(D) ≥ 2 ; h0(L−D) ≥ 2 ; D .(L−D) ≤ k+ 1 ; L+KS ≥ D;
and L ≥ 2D if L2 ≥ 4k+ 3. (†)
One of the main results in [11], which we will use in the proof of the next proposition, is the following:
Lemma 3.7 LetLbe a nef line bundle on a Del Pezzo surface withg(L)≥2andk≥1an integer. If there is a line bundleDsatisfying(†), then there is a smooth curve in|L|of gonality≤k+ 1. If furthermorek≥2and D ∼ −KS8, then all the smooth curves in|L|have gonality≤k+ 1.
Conversely, if there is a smooth curveC∈ |L|of gonalityk+ 1withAag1k+1on it, then there exists a line bundleDsatisfying(†)and such thatOC(D)≥A.
P r o o f. The two first assertions follow from [11, Prop. 3.9 and Prop. 4.1]. (The latter proposition is repro- duced in Proposition 4.1 below.) The last assertion follows from [11, Prop. 3.5].
Also recall the following example of Serrano, which will be useful in the proof of the next proposition:
Example 3.8 (Serrano [21]) LetS be a Del Pezzo surface of degree1. ConsiderL:=−nKS for an integer n≥3and letCbe a smooth curve in|L|. IfCpasses through the base pointxof| −KS|, thenOC(−KS−x) is a base point freegn1−1onC, otherwiseOC(−KS)is a base point freegn1 onC. Serrano shows that in both cases, these pencils compute the gonality and also the Clifford index (for more details see [17, Ex. (2.2)] and [21, Ex. (3.15)]).
We can now prove the following important result, whose proof is unfortunately a bit long and tedious and involves the treatment of several special cases.
Proposition 3.9 Given the assumptions(∗)and the exact sequence(3.6). AssumeC0is exceptional and that Acomputes the Clifford dimension ofC0.
Thenh0(R⊗ωS) = 0.
P r o o f. Assume thath0(R⊗ωS)>0. Then we can putRin a suitable exact sequence as in (3.6), namely
0 −→ P −→ R −→ R1 −→ τ −→ 0, (3.13)
wherePis a line bundle such thatP+KS ≥0,R1is locally free of rankrkR−1≥1andτis of finite length.
ClearlyR1is globally generated off a finite set. SetQ:= detR1. Lemma 3.4 applies forQ, which means that h1(Q) =h2(Q) = 0,Q >0and eitherQ∼ −KS8orQis base point free.
LetC∈ |L|. We get from
0 −→ −M−P −→ Q −→ QC −→ 0 (3.14)
that
h0(QC) = h0(Q) +δ , (3.15)
where we define
δ := h1(−M −P) = h1(L−Q+KS). (3.16)
Sinceh0(Q)≥2andh0(L−Q+KS) =h0(M+P+KS)≥2 (the latter follows fromM+P ≥ −2KS), we have by Lemma 3.6 thatQCcontributes to the Clifford index of anyC∈ |L|.
From (3.6) and (3.13) we get
c2(E) = D . M+c2(R) +lengthτ , (3.17)
c2(R) = P . Q+c2(R1) +lengthτ. (3.18)
Case I:rkR1= 1.
We haveQ=R1andrkE=h0(A) = 3. Furthermore,
CliffQC = CliffA+ 4 +κ , (3.19)
where
κ := −P . M+Q . KS−lengthτ−lengthτ−2δ . (3.20)
Indeed,
CliffQC = Q . L−2
h0(QC)−1
= Q . L−2
h0(Q)−1
−2δ (by (3.15))
= Q . L−Q .(Q−KS)−2δ (sinceh1(Q) = 0)
= Q .(M+P+KS)−2δ
= Q .(M+KS) +c2(E)−D . M−lengthτ−lengthτ−2δ (by (3.17) and (3.18))
= −P . M+Q . KS+ degA−lengthτ−lengthτ−2δ (by (2.5))
= CliffA+ 4−P . M+Q . KS−lengthτ−lengthτ−2δ . SinceAcomputes the Clifford dimension ofC0, we must have
κ ≥ −4. (3.21)
We now divide the proof in several subcases.
Case I–a:P . M = 3.
By (3.19) – (3.21) and (P6) we haveQ∼ −KS8,δ= 0andCliffQC0 = CliffA. Sinceh0 QC0
=h0(Q) = 2, the curveC0is not exceptional, a contradiction.
Case I–b:P . M = 2,Q . KS =−1.
By (3.19) – (3.21) and (P6) we haveQ∼ −KS8,δ= 0andCliffQC0 ≤CliffA+ 1. As above, sinceC0is exceptional, we must have equality in the last inequality, whence by (3.20) we have lengthτ =lengthτ = 0.
SinceQhas one base pointx, it follows thatRis not globally generated by (3.13), whence by (3.6) and Lemma 2.3, we have thatC0passes throughx.
Define the line bundle A := QC0 −xon C0. Then clearly A contributes to the Clifford index of C0, h0(A) = 2andCliffA= CliffA, again contradicting thatC0is exceptional.
Case I–c: P . M = 2,Q . KS =−2. By (3.19) – (3.21) we haveδ=lengthτ =lengthτ = 0,h0(QC) = h0(Q)andCliffA= CliffQC0=Q .(M+P+KS)(the last equality follows by(∗)and Lemma 3.6).
It is clear thatOC0(P)contributes to the Clifford index ofC0, and by Lemma 3.6:
CliffOC0(P) ≤ P .(M+Q+KS)
= 2 +P . Q+P . KS
= CliffA−Q . M−Q . KS+P . KS+ 2
= CliffA−Q . M+P . KS+ 4.
(3.22)
The same reasoning works forM, so we get
Q . M ≤ P . KS+ 4 and Q . P ≤ M . KS+ 4. (3.23) From (3.17) and (3.18), combined withdegA≤g(C0)−1, we get
M .(M +KS) +P . P+KS
+Q2 ≥ 2. (3.24)
Furthermore, sinceQC0computes the Clifford index ofC0andC0is exceptional we must have3≤h0 QC0
= h0(Q) = 12Q .(Q−KS) + 1 = 12Q2+ 2, whence by the Hodge index theorem onQand−KS :
Q2 =
2 if S = S8 or Q∼ −KS7,
4 if Q ∼ −2KS8. (3.25)
IfP2≤0, thenP . KS ≤ −2 (by (P6), sinceP ≥ −KS), and by (3.23) we haveQ . M ≤2. IfQ . M = 2, then CliffOC0(P) ≤ CliffA by (3.22) andh0
OC0(P)
= h0(P) = 2 by Lemma 3.6, contradicting the exceptionality of C0 again. So Q . M ≤ 1 and by the Hodge index theorem and (3.25)M2 ≤ 0. But then M . KS ≤ −2andM .(L−M +KS) = M .(P +Q+KS) ≤2 + 1−2 = 1, whence by Lemma 3.6 and
the fact thatC0is exceptional, we must haveQ . M= 1,M . KS =−2,M2= 0,h0
OC0(M)
=h0(M)and h1(M) = 0. But thenh0
OC0(M)
= 2andOC0(M)computes the Clifford index1ofC0, contradicting the exceptionality ofC0again.
The same reasoning works ifM2≤0.
This means thatP2, M2≥1.
If P ∼ −KS8, then CliffA = Q .(M +P +KS) = Q . M, whence Q . M ≥ 1. At the same time, M .(P+Q) = 2 +M . Q, and one can show thatM satisfies the conditions(†)fork= 1 +M . Q≥2. Since P . M =−KS8. M = 2, we haveM ∼ −KS8, whence by Lemma 3.7 all the smooth curves in|L|have gonality
≤2 +M . Q= CliffA+ 2, whenceC0is not exceptional, a contradiction.
The same reasoning works ifM ∼ −KS8.
SoP2, M2≥1andP . KS, M . KS ≤ −2. By (3.23) we also haveQ . M, Q . P ≤2.
We then easily see by the Hodge index theorem that the only solution to (3.24) and (3.25) isM ∼P ∼Qand M2 =−M . KS = 2. By (3.22) and Lemma 3.6 we getCliffA= CliffOC0(M) =M .(2M +KS) = 2. At the same timeM .(L−M) = 2M2= 4, soM satisfies the conditions(†)fork= 3. Again by Lemma 3.7, all the smooth curves in|L|have gonality≤4, whenceC0is not exceptional, a contradiction.
Case I–d:P . M ≤1.
By combining Lemma 3.6 with (3.19) – (3.21) and (P6), and the fact thath1(Q) = 0, we get as above CliffOC0(P) ≤ CliffA+ 4−Q . M+P . KS, (3.26) whence as above
Q . M ≤ P . KS+ 4 and Q . P ≤ M . KS+ 4. (3.27) As in case I–c we can show thatQ . M ≤ 1if P2 ≤ 0. HenceM .(P +Q+KS) ≤ 2 +M . KS, and by Lemma 3.6 and the fact thatC0 is exceptional, we haveM . KS = −1andP . M = Q . M = 1, whence M ∼ P ∼ Q ∼ −KS8 by (P6) andL ∼ −3KS8, which belongs to the special case studied by Serrano and described in Example 3.8, where none of the smooth curves in|L|are exceptional, a contradiction.
The same reasoning works ifM2 ≤ 0, whenceP2, M2 ≥1, as in case I–c. By the Hodge index theorem M ∼P andM2= 1. By (3.27) we haveQ . M≤3. If equality occurs, thenM ∼ −KS8andCliffOC0(M) = CliffAby (3.26). By Lemma 3.6h0
OC0(M)
=h0(M) = 2, contradicting the exceptionality ofC0. IfQ . M= 2, then by (3.27) we have−M . KS ≤2, whenceM ∼P ∼ −KS8and by (3.26) again
CliffOC0
−KS8
≤ CliffA+ 1. (3.28)
Since−KS8. L=−KS8.
−2KS8+Q
= 2 + 2 = 4,−KS8will induce ag41on every member of|L|, whenceCliffA = 1andC0is a smooth plane quintic, sodegA= 5andg(L) = 6. Moreover, from (3.19) and (3.20) we see thatδ= 0.
We compute
10 = L .
L+KS8
=
−2KS8+Q .
−KS8+Q
= 8 +Q2, whenceQ2= 2. In particular, by Lemma 3.4,Qis base point free, and so isQC0.
Nowg(Q0) = 1, whenceQ≥ −KS8and we have Q ∼ −KS8+ Γ,
for a(−1)–curveΓ. So
L ∼ −3KS8+ Γ.
SinceC0is a smooth plane quintic, it is well–known (see e.g. [14, Beispiel 4]) that we must have 2A ∼ ωC0 ∼
L+KS8
C0 ∼
−2KS8+ Γ
C0. (3.29)
Furthermore, it is classically known (see e.g. [8]) that any pencil computing the gonality ofC0 is obtained by projecting from one point of the curve embedded byA. In other words
OC0
−KS8
∼ A−y , (3.30)
for a pointy∈C0. Combining (3.29) and (3.30) we getOC0(Γ)∼2y, and we have A ∼ OC0
−KS8
+y ∼ QC0−y .
Sinceh1(Q) = 0,h1
−2KS8+ Γ
= 0andδ= 0, we get from (3.15) and (3.16) that h0
QC0
= 1 2Q .
Q−KS8
+ 1 = 3 = h0(A),
contradicting the base point freeness ofQC0.
IfQ . M ≤1, then eitherP ∼ M ∼ Q∼ −KS8 orM . KS ≤ −3. In the first case we getL ∼ −3KS8, which belongs to the special case studied by Serrano and described in Example 3.8, where none of the smooth curves in|L|are exceptional, a contradiction. In the second case, we get by Lemma 3.6
CliffOC0(M) ≤ M .(M+Q+KS) ≤ 1 + 1−3 = −1, a contradiction by Clifford’s theorem.
Case II:rkR1≥2.
We haveh0(A)≥4.
By (2.5), (2.6), (3.17) and (3.18)
CliffA = c2(E)−2(rkE−1)
= D . M +c2(R) +lengthτ−2rkR
= M . P+M . Q+P . Q+c2(R1)−2rkR1−2 +lengthτ+lengthτ.
(3.31)
We can prove by (3.14) – (3.16) that
CliffQC = Q .(M +P) +Q . KS−2δ . (3.32) By using Corollary 3.3 and arguing as in the proof of Proposition 3.5, we get the two possibilities:
Q ∼ −KS8, CliffA ≥ P . M+ CliffQC−2 + 2δ+lengthτ+lengthτ, (3.33) Q ∼ −KS8, CliffA ≥ P . M+ CliffQC−3 + 2δ+lengthτ+lengthτ. (3.34) Note that ifQ2>0, then equality in (3.33) occurs only ifc2(R1)−2rkR1=−Q . KS.
We see that we already get the desired contradictions ifP . M >2in (3.33) and ifP . M >3in (3.34). Also, ifP . M = 3in (3.34), we getCliffA = CliffQC0 andh0
QC0
=h0(Q) =h0
−KS8
= 2, contradicting the exceptionality ofC0.
So we will from now on assume thatP . M ≤2.
Before we divide the rest of the proof in four special cases, we deduce the following important information:
h0(M+P +KS) = 1
2M .(M +KS) +1
2P .(P+KS) +M . P+ 1 +δ , (3.35)
M .(M +KS) ≤ 0 and P .(P+KS) ≤ 0. (3.36)
Indeed, (3.35) follows by Riemann–Roch. For (3.36), assume now, to get a contradiction, that M . (M +KS) ≥ 2. Then h0(M +KS) ≥ 2and by Lemma 3.6, (P +Q)C0 contributes to the Clifford index ofC0, and
Cliff (P+Q)C ≤ (P+Q).(M +KS)
= Q .(M+P+KS)−Q . P+P .(M+KS)
= CliffQC+ 2δ+M . P −Q . P+P . KS.
