Exceptional curves on Del Pezzo surfaces

Math. Nachr. 256, 58 – 81 (2003) / DOI 10.1002/mana.200310070
<strong>Exceptional</strong> <strong>curves</strong> on Del Pezzo surfaces
Andreas Leopold Knutsen ∗1
1 UniversitätEssen,Fachbereich6–Mathematik,45117 Essen, Germany
1 Introduction
Received 3 September 2001, revised 27 June 2002, accepted 2 August 2002
Published online 25 June 2003
Key words Line bundles, <strong>curves</strong>, Del Pezzo surfaces, exceptional <strong>curves</strong>, gonality, Clifford index
MSC (2000) Primary: 14J26; Secondary: 14H51.
We classify all cases of exceptional <strong>curves</strong> on Del Pezzo surfaces, which turn out to be the smooth plane <strong>curves</strong>
and some other cases with Clifford dimension 3. Moreover, the property of being exceptional holds for all
<strong>curves</strong> 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 <strong>curves</strong> in a complete linear system on Del Pezzo surfaces of
degrees ≥ 2 to the case of Del Pezzo surfaces of degree 1, where we explicitly classify the cases where the
gonality and Clifford index are not constant.
The two main purposes of this paper are (1) to classify all cases of exceptional <strong>curves</strong> on Del Pezzo surfaces and
(2) to study the gonality and Clifford index of linearly equivalent smooth <strong>curves</strong> in |L| on a Del Pezzo surface of
degree 1.
In the past decades, several authors have studied the question whether exceptional linear series on a curve C
on a certain surface propagate to the other members of |C|. For instance, for a K3 surface, Saint–Donat proved in
if and only if every smooth curve in |C| does, and Reid [19] extended this result
[20] that C possesses a g1 2 or a g1 3
to other g1 d
s. Harris and Mumford conjectured that linearly equivalent smooth <strong>curves</strong> on a K3 surface 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 <strong>curves</strong> on a K3 surface 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 if OS(C) is ample.
The results of Green and Lazarsfeld were extended to Del Pezzo surfaces of degree ≥ 2 by Pareschi in [17].
In fact, he showed that for deg S ≥ 2 the Clifford index of the smooth <strong>curves</strong> in a linear system |L| containing a
smooth curve of genus g ≥ 4 (which is equivalent to L being nef ) is constant, and that the gonality also is when
g ≥ 2, except for the following case [17, Example (2.1)]:
Case (I): deg S =2and L ∼−2KS (in particular g(L) =3). In this case there is a codimension 1 family of
smooth hyperelliptic <strong>curves</strong> in |L|, whereas the general smooth curve is trigonal.
More precisely, denote by φ : S → P2 the morphism defined by −KS, which is a double cover ramified along
a smooth quartic. Then
H 0 (L) = φ ∗ H 0� O P 2(2) � ⊕ W,
where W is the 1–dimensional subspace of sections vanishing on the ramification locus. The smooth <strong>curves</strong>
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: andreas.knutsen@uni-essen.de
c○ 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 0025-584X/03/25607-0058 $ 17.50+.50/0

Math. Nachr. 256 (2003) 59
By work of Serrano [21] there is an example when deg S =1(namely L ∼−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 for deg S =1.
In this paper we first study exceptional <strong>curves</strong> on Del Pezzo surfaces of any degree. These are the <strong>curves</strong>
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 <strong>curves</strong> are smooth plane <strong>curves</strong>,
and certain <strong>curves</strong> of degree 4r − 3 and genus 4r − 2 in P r for r ≥ 3. We show (Theorem 1.1 (c) below) that
exceptional <strong>curves</strong> on a Del Pezzo surface are exactly the strict transforms of smooth plane <strong>curves</strong> (whence of
Clifford dimension 2), and of smooth <strong>curves</strong> in |−3KS| for deg S =3(of Clifford dimension 3). This is in full
accordance with the conjecture just mentioned. Furthermore, we show that if C is an exceptional curve and A a
line bundle on C computing its Clifford dimension, then A = OC(D) for a line bundle D on the surface and any
smooth curve C ′ ∈|C| is exceptional with its Clifford dimension computed by OC ′(D).
As an application of this study, we classify all the cases for deg S =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 when deg S =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): deg S =1and L ∼−2KS +2Γ,fora(−1)–curve Γ(in particular g(L) =3). In this case there is
also a codimension 1 family of smooth hyperelliptic <strong>curves</strong> in |L|, whereas the general smooth curve is trigonal.
Case (III): deg S =1, L is ample, L.Γ ≥ 2 for all (−1)–<strong>curves</strong> Γ if L 2 ≥ 8, and there is an integer d ≥ 3
such that L 2 ≥ 5d − 8 ≥ 7, L.KS = −d and L.C ≥ d for all smooth rational <strong>curves</strong> C with C 2 =0.
In this case there is a codimension 1 family of |L| of smooth (d − 1)–gonal <strong>curves</strong> (which are exactly the
smooth <strong>curves</strong> passing through the base point of KS), whereas the general smooth curve is d–gonal.
If g(L) ≥ 4, then the smooth <strong>curves</strong> in the codimension 1 family have Clifford index d − 3, whereas the
general smooth curve has Clifford index d − 2.
The three easiest examples of Case (III) for high genus are L ∼−dKS for d ≥ 3 (precisely the case described
by Serrano [21], see Example 3.8 below), L ∼−(d − 1)KS +Γ,ford ≥ 3 and a (−1)–curve Γ, andL ∼
−(d − 2)KS + R,ford ≥ 4 and a smooth rational curve R with R 2 =0. In the case of lowest genus, g(L) =3
(and L 2 =7)and L ∼−2KS +Γfor a (−1)–curve Γ.
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 of Pic S in the representation of L. 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, if L is a nef
line bundle on a Del Pezzo surface, we define the (finite) set of <strong>curves</strong> R(L) by:
R(L) := {Γ | Γ=(−1)–curve and Γ.L=0} ,
and denote its cardinality by m(L). (Note that by the Hodge index theorem, the <strong>curves</strong> in R(L) are necessarily
disjoint.) Moreover, see Section 2.3 for the definition of Sn.
Theorem 1.1 Let L be a nef line bundle on a Del Pezzo surface S with g(L) ≥ 2.
(a) All the smooth <strong>curves</strong> in |L| have the same gonality if and only if L is not as in the cases (I), (II) or (III)
above.
(b) If g(L) ≥ 4, then all the smooth <strong>curves</strong> in |L| have the same Clifford index if and only if L is not as in case
(III) above.
(c) If g(L) ≥ 4, then either all or none of the smooth <strong>curves</strong> in |L| are exceptional, and if they are, they all have
the same Clifford index and Clifford dimension. Furthermore, the <strong>curves</strong> are exceptional of Clifford dimension
r ≥ 2 and Clifford index c if and only if L is as in one of the two following cases:
(i) There exists a base point free line bundle D on S satisfying
D 2 = 1, D.KS = −3 , D.L = c +4 and Γ.L ≤ 1 for any Γ ∈R(D)
(the latter condition is equivalent to DC being very ample for all C ∈|L|).
In this case r =2, the Clifford dimension is computed only by OC(D) for all smooth <strong>curves</strong> C ∈|L| and
φD : S → P 2 gives the identification S � S m(D) and maps every smooth curve in |L| to a smooth plane curve
of degree D.L= c +4, whence all these <strong>curves</strong> are exceptional of Clifford dimension 2 and Clifford index c.

60 Knutsen: <strong>Exceptional</strong> <strong>curves</strong> on Del Pezzo surfaces
(ii) S � Sn for n ∈{6, 7, 8} and
n−6 �
L ∼ −3KSn + aiΓi , Γi = (−1)–curve , ai∈{2, 3} .
i=1
In this case r =3, g(L) =10, c =3and for all smooth <strong>curves</strong> C ∈|L|, the Clifford dimension is computed
only by OC(D), forD := −KSn + �n−6 i=1 Γi.
We actually prove a more precise result. In fact, any line bundle A computing 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 of S is � equivalently, the more points of P2 we blow up � , then the more “special” examples we get. In fact, up to deg S =4, all linearly equivalent smooth
<strong>curves</strong> have the same gonality and Clifford index and the only exceptional <strong>curves</strong> are the smooth plane <strong>curves</strong>. For
deg S =3, new examples of exceptional <strong>curves</strong> turn up, for deg S =2, the first example of linearly equivalent
smooth <strong>curves</strong> of different gonalities appears, and for deg S =1, more such examples turn up, together with
examples of linearly equivalent smooth <strong>curves</strong> of different Clifford indices.
It would be an interesting problem, to study the picture when one blows up ≥ 9 points of P2 (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 of P2 .
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 <strong>curves</strong> 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 degree 1, we need a more careful analysis. We also use results about exceptional <strong>curves</strong> 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 for deg S ≥ 2 are equal to the results of Pareschi [17]. In our
proofs, we however never assume that deg S =1, both to give a more complete exposition, and also because
restricting to the case deg S =1would not make the proofs easier, since the case deg S =1is exactly the case
where most problems show up.
Remark 1.2 The smooth <strong>curves</strong> in case (II) above are simply the strict transforms of smooth <strong>curves</strong> in |−2KS|
for deg S =2which do not pass through the blown up point.
The strict transforms of such <strong>curves</strong> passing through the blown up point are the <strong>curves</strong> 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 are smooth irreducible algebraic surfaces.
By BP1,...,Pn(S) we will mean the blowing up of S at the points P1,...,Pn.
By a curve on a surface S is always meant a reduced 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 ∼.
If L is any line bundle on a surface, L is said to be numerically effective, orsimplynef, ifL.C ≥ 0 for all
<strong>curves</strong> C on S. In this case L is said to be big if L2 > 0. Moreover we write L ≥ 0 for L effective and L>0 for
L effective and nonzero.
If F is any coherent sheaf on a variety V , we shall denote by hi (F) the complex dimension of Hi (V,F), and
by χ(F) the Euler characteristic � (−1) ihi (F).
The sectional genus of a line bundle L is the integer g(L) = 1
2 L.(L + KS)+1.IfD∈|L|,theng(L) is the
arithmetic genus of D.

Math. Nachr. 256 (2003) 61
If C is a curve on a surface and L is a line bundle on the surface, we often use the notation LC := L ⊗OC.
2.2 k–very ampleness
A line bundle L on a smooth variety X is said to be k–very ample (resp. k–spanned), for an integer k ≥ 0, if
for each (resp. each curvilinear) 0–dimensional subscheme Z of X of length h0� �
OZ = k +1, the restriction
map H0 (L) → H0� �
L⊗OZ is surjective. (Recall that a 0–dimensional scheme Z is called curvilinear if
dim TxZ ≤1 for every x ∈Zred.)
In [12] we made the following definition: A line bundle L on a smooth surface S is birationally k–very ample
(resp. birationally k–spanned), for an integer k ≥ 1, if there exists a non–empty Zariski–open subset U of S
such that the restriction map H0 (L) → H0� �
L ⊗OZ is surjective for any (resp. any curvilinear) 0–dimensional
subscheme Z of S of length h0� �
OZ = k +1with support in U.
2.3 Del Pezzo surfaces
A surface S is called a Del Pezzo surface if its anticanonical bundle −KS is ample. The degree of S is defined
as deg S = K2 S . We have the following classification of such surfaces:
Proposition 2.1 [4] Let S be a Del Pezzo surface. Then
(a) 1 ≤ deg S ≤ 9,
(b) deg S =9if and only if S � P2 ,
(c) if deg S =8, then either S � P1 × P1 �
2 or S � BP P � ,
�
2 (d) if 1 ≤ deg S ≤ 7,thenS�BP1,...,P9−deg P S
� .
We will often denote a Del Pezzo surface of degree deg S by Sn, wheren =9− deg S. Letπ : S → P2 be
the blowing up as in (d). Denote by l the class of π∗�OP2(1) � and by ei the class of π−1 (Pi). Wethenhave
and
l 2 = 1, ei .ej = −δij , ei .l = 0,
Pic Sn � Zl ⊕ Ze1 ⊕ ...⊕ Zen � Z n+1 .
Therefore any line bundle on Sn is of the form al − � n
1 biei and −KS =3l − � n
1 ei.
A (−1)–curve on a Del Pezzo surface is a curve Γ satisfying Γ 2 = −1. Such a curve is necessarily smooth
and rational and satisfies Γ .KS = −1 (by the adjunction formula), and those are the only <strong>curves</strong> with negative
self–intersection on a Del Pezzo surface. It follows that a line bundle D ≥ 0 is not nef if and only if there exists
a (−1)–curve Γ such that Γ .D 0, L �∼ −KS8 and L is not a (−1)–curve on S, then−L.KS ≥ 2 (Hodge index theorem and
adjunction formula).

62 Knutsen: <strong>Exceptional</strong> <strong>curves</strong> on Del Pezzo surfaces
(P7) If L is nef with L 2 > 0 we have h i (L) =h i (L + KS) =0,fori =1, 2, h 0 (L + KS) =g(L) ≥ 1 and the
general member of |L| is smooth and irreducible.
To show the last condition, note that h 1 (L) =h 2 (L) =0by (P5) and h 2 (L + KS) =h 0 (−L) =0by Serre
duality since L>0. Moreover, L is numerically 1–connected by the Hodge index theorem � since L 2 > 0 � and
so h 1 (L + KS) =0by Ramanujam vanishing. By (P1) and (P4) L is 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 bundles E(C, A)
Let C be a smooth irreducible curve of genus g ≥ 2. We denote by g r d
d. The gonality gon C of C is defined as
gon C := min � k | C has a g 1� k .
a linear system of dimension r and degree
In particular, C is called hyperelliptic if gon C =2and trigonal if gon C =3. Note that if gon C = k, allg 1 k s
must necessarily be base point free and complete.
If g ≥ 4 and A is a line bundle on C, then the Clifford index of A is the integer
Cliff A := deg A − 2 � h 0 (A) − 1 �
and the Clifford index of C itself is defined as
Cliff C := min � Cliff A | h 0 (A) ≥ 2, h 1 (A) ≥ 2 � .
Clifford’s theorem then states that Cliff C ≥ 0 with equality if and only if C is hyperelliptic and Cliff C =1if
and only if C is trigonal or a smooth plane quintic.
⌊ g+3
2
At the other extreme, we get from Brill–Noether theory (cf. [1, V]) that the gonality of C satisfies gon C ≤
⌋, whence Cliff C ≤⌊g−1
2
⌋. For the general curve of genus g,wehaveCliff C = ⌊ g−1
2 ⌋.
We say that a line bundle A on C contributes to the Clifford index of C if both h 0 (A) ≥ 2 and h 1 (A) ≥ 2,
and that it computes the Clifford index of C if in addition Cliff C = Cliff A.
Note that Cliff A = Cliff � ωC ⊗ A −1� and also observe that by Riemann–Roch
h 0 (A)+h 1 (A) = g +1− Cliff A. (2.1)
The Clifford dimension of C is defined as
min � h 0 (A) − 1 | A computes the Clifford index of C � .
A line bundle A which achieves the minimum and computes the Clifford index, is said to compute the Clifford
dimension. A curve of Clifford index c is (c +2)–gonal if and only if it has Clifford dimension 1. For a general
curve C, we have gon C = c +2.
Lemma 2.2 [3, Theorem 2.3] The gonality k of a smooth (irreducible) projective curve C satisfies
Cliff C +2 ≤ k ≤ Cliff C +3.
The <strong>curves</strong> satisfying gon C = Cliff C +3are conjectured to be very rare and called exceptional (cf. [8]
or Proposition 3.11 and Remark 3.12 below). The <strong>curves</strong> of Clifford dimension 2 are exactly the smooth plane
<strong>curves</strong> of degree ≥ 5.
Let C be a smooth curve on a regular surface S � i.e. h 1 (OS) =0 � . Recall that if A is a line bundle on C
which is generated by its global sections, then one can associate to the pair (C, A) a vector bundle E(C, A) of
rank h 0 (A) as follows (this vector bundle was introduced by Lazarsfeld in [13], for more details we refer to [2]
and [17]). Thinking of A as a coherent sheaf on S, we get a short exact sequence
0 −→ F (C, A) −→ H 0 (A) ⊗C OS −→ A −→ 0 (2.2)

Math. Nachr. 256 (2003) 63
of OS–modules, where F (C, A) is locally free (since A is locally isomorphic to OC and hence has homological
dimension one over OS).
Dualizing this sequence, one gets
0 −→ H 0 (A) ∨ ⊗OS −→ E(C, A) −→ N C/S ⊗ A ∨ −→ 0 , (2.3)
where E(C, A) :=F (C, A) ∨ . This vector bundle has the following properties:
det E(C, A) = OS(C) , (2.4)
c2(E(C, A)) = deg A, (2.5)
rk E(C, A) = h 0 (A) ,
h
(2.6)
i (E(C, A) ∨ ) = h 2−i� �
E(C, A) ⊗ ωS = 0,
h
i = 0, 1 , (2.7)
0 (E(C, A)) = h 0 (A)+h 0� NC/S ⊗ A ∨� , (2.8)
If A ≤NC/S, thenE(C, A) is generated by its global sections off a finite
set coinciding with the (possibly zero) base divisor of NC/S ⊗ A
(2.9)
∨ .
Note that if A is any line bundle on C computing the gonality or the Clifford index of C, thenAisgenerated by its global sections, and we can carry out the construction of the vector bundle E(C, A) above.
Specializing to Del Pezzo surfaces, we get from (2.3) tensored by ωS that
h 1 (A) = h 0� �
E(C, A) ⊗ ωS . (2.10)
Moreover, we have NC/S ⊗ A∨ � ωC ⊗ A∨ ⊗OC(−KS), and by (P1) we get:
Lemma 2.3 Assume A and ωC ⊗ A∨ are both base point free (as will be the case if A computes the Clifford
index of C).
Then E(C, A) is generated by its global sections except possibly at the base point x of |−KS|,whenK2 S =1
and C passes through x.
3 <strong>Exceptional</strong> <strong>curves</strong>
We will in this section study the exceptional <strong>curves</strong> on a Del Pezzo surface. Clearly, on a Del Pezzo surface
of type Sn, all the strict transforms of the smooth plane <strong>curves</strong> are exceptional. The following is a non–trivial
example of smooth <strong>curves</strong> of Clifford dimension 3.
Example 3.1 Let S � S6 and consider the line bundle L ∼−3KS6. All the smooth <strong>curves</strong> in |L| are of genus
10,andforanysuchC∈|L|,wehaveh0� � ��
1
OC − KS6 = h � � �� � �
OC − KS6 =4,soOC − KS6 contributes
to the Clifford index of C. We compute
� � � � � 0
Cliff OC − KS6 = degOC−KS6
− 2 h � � �� �
OC − KS6 − 1 = 3.
Now assume C is not exceptional of Clifford index 3. Thend := gon C ≤ 5, and by [11, Prop. 3.7] there has
to exist an effective divisor D satisfying D.L− D 2 = d andsuchthatD is of one of the following types:
(a) D 2 =0, −D.KS6 =2,
(b) D 2 =1, −D.KS6 =3,
(c) D ∼−KS6.
In these three cases we get respectively D.L− D 2 = −3D.KS6 − D 2 =6, 8, 6, whence a contradiction.
Hence all the smooth <strong>curves</strong> in |L| have gonality 6 and Clifford index 3, which shows that they are exceptional.
If the Clifford dimension of any such C is 2, thenC is isomorphic to a smooth plane septic, and we easily get a
contradiction from the genus formula of a smooth plane curve. So all the <strong>curves</strong> have Clifford dimension 3.
We show in this section that the only exceptional <strong>curves</strong> on a Del Pezzo surface are precisely the strict transforms
of the smooth plane <strong>curves</strong> and of the <strong>curves</strong> in the above example.
First we need the following result, which we formulate in a general setting:

64 Knutsen: <strong>Exceptional</strong> <strong>curves</strong> on Del Pezzo surfaces
Proposition 3.2 Let R be a vector bundle of rank ≥ 2 on a surface S generated by its global sections off a
finite set and such that H 0 (R ∨ )=0and c1(R) 2 ≥ 0.
(a) If c1(R) 2 > 0, thenc2(R) − 2rk R ≥ min{−2,c1(R) .ωS}. Iffurthermoreh 0 (ωS) =h 1 (ωS) =0and
h 0 (R ⊗ ωS) > 0,thenc2(R) − 2rk R ≥−2.
(b) If c1(R) 2 =0and h 1 (OS) =0,thendet R ∼OS(kΣ) for a base point free pencil |Σ| on S and an integer
k ≥ 1, and c2(R) − 2rk R ≥ k min{−2, Σ .ωS}.
Proof. Let r := rk R. For a general r–dimensional subspace V ⊆ H 0 (R), the evaluation map eV :
V ⊗OS → R is generically an isomorphism and has rank exactly r − 1 along a divisor D ∈|det R| defined
by the vanishing of det eV . By Bertini’s theorem, D is smooth except possibly at the points where R fails to be
generated. Moreover, B := coker eV is a torsion free sheaf on D (which is a line bundle outside of the singular
points of D), and we have the exact sequence:
0 −→ V ⊗OS −→ R −→ B −→ 0 (3.1)
of OS–modules. Dualizing this sequence, one obtains
0 −→ R ∨ −→ V ∨ ⊗OS −→ A −→ 0 , (3.2)
�
where A := Hom OD B,OD(D) � is also torsion free. Moreover we find from (3.1) and (3.2) that
c2(R) = degA, (3.3)
rk R ≤ h 0 (A) , (3.4)
deg B =2pa(D) − 2 − deg A − D.KS , (3.5)
where the degree of A is defined by the Riemann–Roch type formula deg A := χ(A) − χ(OD) =χ(A) +
pa(D) − 1, and similarly for B.
To prove (a), let now D 2 > 0. Then by Bertini’s theorem, D is reduced and irreducible (i.e. what we call a
curve).
If h 1 (A) > 0, then by Clifford’s theorem for torsion free rank 1 sheaves on singular <strong>curves</strong> (see the appendix
of [7]), Cliff A ≥ 0, whence by (3.3) and (3.4)
c2(R) − 2rk R ≥ deg A − 2h 0 (A) ≥ Cliff A − 2 ≥ −2 ,
which is what we want to prove. Also note that h 1 (A) =h 0 (R ⊗ ωS) when h 0 (ωS) =h 1 (ωS) =0.
If h 1 (A) =0,thendeg A = h 0 (A)+pa(D) − 1. By (3.1) B is generated by its global sections off a finite
set, whence h 0 (B) > 0 and deg B ≥ 0. Combining (3.3), (3.4) and (3.5) we get
c2(R) − 2rk R ≥ deg A − 2h 0 (A) = − deg A − 2+2pa(D) = degB + D.KS ≥ D.KS ,
which finishes the proof of (a).
Finally we prove (b), so let D 2 =0. Since the evident map Λ r H 0 (R) → Λ r R = OS(D) is surjective off
a finite set, |D| is a linear system without fixed components, and is therefore base point free, since D 2 =0.
Since we assume S to be regular, it follows that |D| is composed with a rational pencil, i.e. there is a smooth
curve Σ ⊆ S and an integer k ≥ 1 such that D ∼ kΣ and D is the disjoint union of k <strong>curves</strong> in |Σ|, write
D = D1 + ...+ Dk.
The sheaves A and B are therefore the direct sums A = ⊕Ai and B = ⊕Bi of sheaves on each Di. By
arguing as in the proof of (a) we find that deg Ai − 2h 0 (Ai) ≥ min{−2,Di .KS}, whence
c2(R) − 2rk R ≥ deg A − 2h 0 (A) = � � deg Ai − 2h 0 (Ai) � ≥ k min{−2, Σ .KS} .
This concludes the proof of (b).
In particular, by specializing to a Del Pezzo surface, using (P6) and the fact that h 0 (kΣ) = k +1by (P5),
we get

Math. Nachr. 256 (2003) 65
Corollary 3.3 With the same assumptions as in Proposition 3.2, assume that S is a Del Pezzo surface. Set
D := det R. Then
(a) c2(R) − 2rk R ≥−2,whenD∼−KS8, (b) c2(R) − 2rk R ≥−2r, forr := h0 (D) − 1,whenD2 =0,
(c) c2(R) − 2rk R ≥ D.KS =2g(D) − D2 − 2 otherwise.
Furthermore, in case (c),ifh0 (R ⊗ ωS) > 0, then we have the stronger inequality c2(R) − 2rk R ≥−2.
We now make the following assumptions:
(∗) Let C0 be a smooth curve of genus g ≥ 4 on a Del Pezzo surface. Set L := OS(C0). LetA be a base point
free line bundle on C0 such that 3 ≤ h 0 (A) ≤ h 1 (A) (this implies deg A ≤ g − 1) and ωC0 ⊗ A ∨ is base
point free.
As in Subsection 2.4 we get the vector bundle E := E(C0,A), which is of rank h 0 (A) ≥ 3. Since by (2.10)
h 0 (E ⊗ ωS) > 0, we can find an invertible subsheaf 0 → M → E, and after saturating, we can assume we have
an exact sequence
0 −→ M −→ E −→ R −→ τ −→ 0 , (3.6)
where R is locally free of rank ≥ 2 and τ is supported on a finite set.
Define D := det R. Wehave
Lemma 3.4 (a) h0 (R∨ )=h1 (R∨ )=0,
(b) D is non trivial and either D ∼−KS8or D is base point free. In particular h1 (D) =h2 (D) =0.
Moreover L ∼ M + D.
P r o o f. Similar to the proof of [17, (1.12)].
For any C ∈|L| we write DC := OC(D).
Proposition 3.5 (a) h0 (DC) is the same for any C ∈|L|,
(b) DC0 ≥ A,
(c) h1 (DC) =h0 (M ⊗ ωS) > 0 for any C ∈|L|,
(d) Cliff DC is the same for any C ∈|L| and Cliff DC ≤ Cliff A.
P r o o f. The statements (a) – (c) follow as in the proof of [17, (1.13)]. To prove (d), note that R is globally
generated off a finite set, so Corollary 3.3 applies.
We have
Indeed,
Cliff A ≥ D.M + c2(R) − 2rk R. (3.7)
Cliff A = c2(E) − 2(rk E − 1) (by (2.5) and (2.6))
= D.M + c2(R)+length τ − 2rk R
≥ D.M + c2(R) − 2rk R.
(by (3.6))
If D 2 =0, then by (3.7) and (b) of Corollary 3.3:
Cliff A ≥ D.L− 2r. (3.8)
At the same time, by the exact sequence
and Lemma 3.4,
0 −→ −M −→ D −→ DC −→ 0 (3.9)
h 0 (DC) = h 0 (D)+h 1 (−M) ≥ r +1. (3.10)

66 Knutsen: <strong>Exceptional</strong> <strong>curves</strong> on Del Pezzo surfaces
Combining (3.8) and (3.10) we get the desired result Cliff A ≥ Cliff DC.
Assume now D2 > 0. Wehave
h 0� � 0 1 0 1
DC = h (D)+h (−M) ≥ h (D) =
2 D.(D − KS)+1,
whence
(3.11)
Cliff DC = D.L− 2 � h 0� � �
DC − 1 ≤ D.M + D.KS . (3.12)
If D ∼−KS8, thenh0 (D) =2, and since DC0 ≥ A and h0 (A) ≥ 3, wehaveh0� �
0
DC = h � we have strict inequalities in (3.11) and (3.12).
�
DC0 ≥ 3, so
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 Let L be a nef line bundle on a Del Pezzo surface with g(L) ≥ 4. Assume L + KS ∼ D1 + D2
for two divisors D1 and D2 satisfying h0 (Di) ≥ 2. ThenOC(D1) and OC(D2) contribute to the Clifford index
of any smooth C ∈|L| and
Cliff OC(D1) = Cliff OC(D2) ≤ D1 .D2 ,
with equality if and only if h 0 (OC(Di)) = h 0 (Di) and h 1 (Di) =0for i =1, 2.
Furthermore, if equality holds and there exists a smooth curve C0 ∈|L| such that Cliff C0 = D1 .D2, then
any (−1)–curve Γ with Γ .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) ≥ 2 and h1 (OC(D1)) ≥ h2 (KS − D2) =h0 (D2) ≥ 2, sothatOC(D1) contributes to
the Clifford index of any smooth C ∈|L|.BysymmetrywegetthesameresultforD2.
Now OC(D2) �OC(L + KS − D1) � ωC ⊗OC(D1) ∨ ,soCliff OC(D1) = Cliff OC(D2).
One computes by Riemann–Roch on S:
Cliff OC(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 if h 0 (OC(Di)) = h 0 (Di) and h 1 (Di) =0for i =1, 2.
For the last statement, since −1 =Γ. (L+KS) =Γ.D1 +Γ.D2, assume by symmetry to get a contradiction
that Γ .D1 > 0. WethengetΓ .D2 = −1−Γ .D1 ≤−2. Clearly h 0 (D1 +Γ) ≥ h 0 (D1) ≥ 2 and h 0 (D2 −Γ) =
h 0 (D2) ≥ 2, so by what we have just shown OC(D1+Γ) contributes to the Clifford index of any smooth C ∈|L|,
and
a contradiction.
Cliff OC0(D1 +Γ) ≤ (D1 +Γ). (D2 − Γ) = D1 .D2 − 2Γ .D1 < Cliff C0 ,
As in [11] we say that a line bundle D satisfies the conditions (†) for an integer k ≥ 1 if:
h 0 (D) ≥ 2; h 0 (L − D) ≥ 2; D.(L − D) ≤ k +1; L + KS ≥ D ;
and L ≥ 2D if L 2 ≥ 4k +3.
One of the main results in [11], which we will use in the proof of the next proposition, is the following:
(†)

Math. Nachr. 256 (2003) 67
Lemma 3.7 Let L be a nef line bundle on a Del Pezzo surface with g(L) ≥ 2 and k ≥ 1 an integer. If there is
a line bundle D satisfying (†), then there is a smooth curve in |L| of gonality ≤ k +1.Iffurthermorek≥2and D �∼ −KS8, then all the smooth <strong>curves</strong> in |L| have gonality ≤ k +1.
Conversely, if there is a smooth curve C ∈|L| of gonality k +1with A a g1 k+1 on it, then there exists a line
bundle D satisfying (†) and such that OC(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 reproduced
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]) Let S be a Del Pezzo surface of degree 1. Consider L := −nKS for an integer
n ≥ 3 and let C be a smooth curve in |L|. IfC passes through the base point x of |−KS|, thenOC(−KS − x)
is a base point free g 1 n−1 on C, otherwiseOC(−KS) is a base point free g 1 n on C. 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). Assume C0 is exceptional and that
A computes the Clifford dimension of C0.
Then h0 (R ⊗ ωS) =0.
Proof. Assumethath 0 (R ⊗ ωS) > 0. Then we can put R in a suitable exact sequence as in (3.6), namely
0 −→ P −→ R −→ R1 −→ τ ′ −→ 0 , (3.13)
where P is a line bundle such that P + KS ≥ 0, R1 is locally free of rank rk R − 1 ≥ 1 and τ ′ is of finite length.
Clearly R1 is globally generated off a finite set. Set Q := det R1. Lemma 3.4 applies for Q, which means that
h 1 (Q) =h 2 (Q) =0, Q>0 and either Q ∼−KS8 or Q is base point free.
Let C ∈|L|. We get from
that
where we define
0 −→ −M − P −→ Q −→ QC −→ 0 (3.14)
h 0 (QC) = h 0 (Q)+δ, (3.15)
δ := h 1 (−M − P ) = h 1 (L − Q + KS) . (3.16)
Since h 0 (Q) ≥ 2 and h 0 (L − Q + KS) =h 0 (M + P + KS) ≥ 2(the latter follows from M + P ≥−2KS),
we have by Lemma 3.6 that QC contributes to the Clifford index of any C ∈|L|.
From (3.6) and (3.13) we get
c2(E) = D.M + c2(R)+length τ, (3.17)
c2(R) = P.Q+ c2(R1)+length τ ′ . (3.18)
Case I: rk R1 =1.
We have Q = R1 and rk E = h 0 (A) =3.Furthermore,
where
Cliff QC = Cliff A +4+κ, (3.19)
κ := −P .M+ Q.KS − length τ − length τ ′ − 2δ. (3.20)

68 Knutsen: <strong>Exceptional</strong> <strong>curves</strong> on Del Pezzo surfaces
Indeed,
Cliff QC = Q.L− 2 � h0 (QC) − 1 �
= Q.L− 2 � h0 (Q) − 1 � − 2δ (by (3.15))
= Q.L− Q.(Q − KS) − 2δ (since h1 = Q.(M + P + KS) − 2δ
(Q) =0)
= Q.(M + KS)+c2(E) − D.M − length τ − length τ ′ − 2δ (by (3.17) and (3.18))
= −P .M+ Q.KS +degA−length τ − length τ ′ − 2δ
= Cliff A +4− P.M+ Q.KS − length τ − length τ
(by (2.5))
′ − 2δ.
Since A computes the Clifford dimension of C0,wemusthave
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 have Q ∼−KS8, δ =0and Cliff QC0 = Cliff A. Sinceh0� �
0
QC0 = h (Q) =
2, thecurveC0 is not exceptional, a contradiction.
Case I–b: P.M=2, Q.KS = −1.
By (3.19) – (3.21) and (P6) we have Q ∼−KS8, δ =0and Cliff QC0 ≤ Cliff A +1. As above, since C0 is
exceptional, we must have equality in the last inequality, whence by (3.20) we have length τ = length τ ′ =0.
Since Q has one base point x, it follows that R is not globally generated by (3.13), whence by (3.6) and Lemma
2.3, we have that C0 passes through x.
Define the line bundle A ′ := QC0 − x on C0. Then clearly A ′ contributes to the Clifford index of C0,
h 0 (A ′ )=2and Cliff A ′ = Cliff A, again contradicting that C0 is exceptional.
Case I–c: P.M =2, Q.KS = −2. By (3.19) – (3.21) we have δ = length τ = length τ ′ =0, h 0 (QC) =
h 0 (Q) and Cliff A = Cliff QC0 = Q.(M + P + KS) (the last equality follows by (∗) and Lemma 3.6).
It is clear that OC0(P ) contributes to the Clifford index of C0, and by Lemma 3.6:
Cliff OC0(P ) ≤ P.(M + Q + KS)
=2+P.Q+ P.KS
= Cliff A − Q.M − Q.KS + P.KS +2
= Cliff A − Q.M + P.KS +4.
The same reasoning works for M,soweget
(3.22)
Q.M ≤ P.KS +4 and Q.P ≤ M.KS +4. (3.23)
From (3.17) and (3.18), combined with deg A ≤ g(C0) − 1,weget
M.(M + KS)+P. � � 2
P + KS + Q ≥ 2 . (3.24)
Furthermore, since QC0 computes the Clifford index of C0 and C0 is exceptional we must have 3 ≤ h0� �
QC0 =
h 0 (Q) = 1
2
Q.(Q − KS)+1= 1
2 Q2 +2, whence by the Hodge index theorem on Q and −KS :
Q 2 =
� 2 if S = S8 or Q ∼−KS7 ,
4 if Q ∼ −2KS8 .
(3.25)
If P 2 ≤ 0, thenP.KS ≤−2(by (P6), since P ≥−KS), and by (3.23) we have Q.M ≤ 2. IfQ.M =2,
then Cliff OC0(P ) ≤ Cliff A by (3.22) and h 0� OC0(P ) � = h 0 (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) M 2 ≤ 0. But then
M.KS ≤−2 and M.(L − M + KS) =M.(P + Q + KS) ≤ 2+1− 2=1, whence by Lemma 3.6 and

Math. Nachr. 256 (2003) 69
the fact that C0 is exceptional, we must have Q.M =1, M.KS = −2, M 2 =0, h 0� OC0(M) � = h 0 (M) and
h 1 (M) =0.Butthenh 0� OC0(M) � =2and OC0(M) computes the Clifford index 1 of C0, contradicting the
exceptionality of C0 again.
The same reasoning works if M 2 ≤ 0.
This means that P 2 ,M 2 ≥ 1.
If P ∼ −KS8, thenCliff A = Q.(M + P + KS) = Q.M, whence Q.M ≥ 1. At the same time,
M.(P + Q) =2+M.Q, and one can show that M satisfies the conditions (†) for k =1+M.Q≥ 2. Since
P.M= −KS8 .M =2,wehaveM �∼ −KS8, whence by Lemma 3.7 all the smooth <strong>curves</strong> in |L| have gonality
≤ 2+M.Q= Cliff A +2, whence C0 is not exceptional, a contradiction.
The same reasoning works if M ∼−KS8.
So P 2 ,M 2 ≥ 1 and P.KS,M.KS ≤−2. By (3.23) we also have Q.M,Q.P ≤ 2.
We then easily see by the Hodge index theorem that the only solution to (3.24) and (3.25) is M ∼ P ∼ Q and
M 2 = −M .KS =2. By (3.22) and Lemma 3.6 we get Cliff A = Cliff OC0(M) =M.(2M + KS) =2.At
the same time M.(L − M) =2M 2 =4,soM satisfies the conditions (†) for k =3. Again by Lemma 3.7, all
the smooth <strong>curves</strong> in |L| have gonality ≤ 4, whence C0 is 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 that h 1 (Q) =0, we get as above
whence as above
Cliff OC0(P ) ≤ Cliff A +4− Q.M + P.KS , (3.26)
Q.M ≤ P.KS +4 and Q.P ≤ M.KS +4. (3.27)
As in case I–c we can show that Q.M ≤ 1 if P 2 ≤ 0. Hence M.(P + Q + KS) ≤ 2+M.KS, and
by Lemma 3.6 and the fact that C0 is exceptional, we have M.KS = −1 and P.M = Q.M =1, whence
M ∼ P ∼ Q ∼−KS8 by (P6) and L ∼−3KS8, which belongs to the special case studied by Serrano and
described in Example 3.8, where none of the smooth <strong>curves</strong> in |L| are exceptional, a contradiction.
The same reasoning works if M 2 ≤ 0, whence P 2 ,M2 ≥ 1, as in case I–c. By the Hodge index theorem
M ∼ P and M 2 =1. By (3.27) we have Q.M ≤ 3. If equality occurs, then M ∼−KS8 and Cliff OC0(M) =
Cliff A by (3.26). By Lemma 3.6 h0�OC0(M) � = h0 (M) =2, contradicting the exceptionality of C0.
If Q.M =2, then by (3.27) we have −M .KS≤ 2, whence M ∼ P ∼−KS8 and by (3.26) again
� �
Cliff OC0 − KS8 ≤ Cliff A +1. (3.28)
Since −KS8 .L = −KS8 . � − 2KS8 + Q � =2+2=4, � �
� − � 1
KS8 will induce a g4 on every member of |L|,
whence Cliff A =1and C0 is a smooth plane quintic, so deg A =5and g(L) =6. Moreover, from (3.19) and
(3.20) we see that δ =0.
We compute
10 = L. � � �
L + KS8 = − 2KS8 + Q � . � − KS8 + Q � = 8+Q 2 ,
whence Q 2 =2. In particular, by Lemma 3.4, Q is base point free, and so is QC0.
Now g(Q0) =1, whence Q ≥−KS8 and we have
Q ∼ −KS8 +Γ,
for a (−1)–curve Γ. So
L ∼ −3KS8 +Γ.
Since C0 is a smooth plane quintic, it is well–known (see e.g. [14, Beispiel 4]) that we must have
2A ∼ ωC0 ∼ � �
L + KS8 ∼ � − 2KS8 +Γ �
. (3.29)
C0
C0

70 Knutsen: <strong>Exceptional</strong> <strong>curves</strong> on Del Pezzo surfaces
Furthermore, it is classically known (see e.g. [8]) that any pencil computing the gonality of C0 is obtained by
projecting from one point of the curve embedded by A. Inotherwords
� �
OC0 − KS8 ∼ A − y, (3.30)
for a point y ∈ C0. Combining (3.29) and (3.30) we get OC0(Γ) ∼ 2y, and we have
� �
A ∼ OC0 − KS8 + y ∼ QC0 − y.
Since h 1 (Q) =0, h 1� − 2KS8 +Γ � =0and δ =0, we get from (3.15) and (3.16) that
h 0� � 1
QC0 =
2 Q.� � 0
Q − KS8 +1 = 3 = h (A) ,
contradicting the base point freeness of QC0.
If Q.M ≤ 1, then either P ∼ M ∼ Q ∼−KS8 or M.KS ≤−3. In the first case we get L ∼−3KS8,
which belongs to the special case studied by Serrano and described in Example 3.8, where none of the smooth
<strong>curves</strong> in |L| are exceptional, a contradiction. In the second case, we get by Lemma 3.6
Cliff OC0(M) ≤ M.(M + Q + KS) ≤ 1+1− 3 = −1 ,
a contradiction by Clifford’s theorem.
Case II: rk R1 ≥ 2.
We have h 0 (A) ≥ 4.
By (2.5), (2.6), (3.17) and (3.18)
Cliff A = c2(E) − 2(rk E − 1)
= D.M + c2(R)+length τ − 2rk R
= M.P+ M.Q+ P.Q+ c2(R1) − 2rk R1 − 2+length τ + length τ ′ .
We can prove by (3.14) – (3.16) that
(3.31)
Cliff QC = 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 , Cliff A ≥ P.M+ Cliff QC − 2+2δ + length τ + length τ ′ , (3.33)
Q ∼ −KS8 , Cliff A ≥ P.M+ Cliff QC − 3+2δ + length τ + length τ ′ . (3.34)
Note that if Q2 > 0, then equality in (3.33) occurs only if c2(R1) − 2rk R1 = −Q.KS.
We see that we already get the desired contradictions if P.M>2 in (3.33) and if P.M>3 in (3.34). Also,
if P.M =3in (3.34), we get Cliff A = Cliff QC0 and h0� �
0 0
QC0 = h (Q) =h � �
− KS8 =2, contradicting
the exceptionality of C0.
So we will from now on assume that P.M≤ 2.
Before we divide the rest of the proof in four special cases, we deduce the following important information:
h 0 (M + P + KS) = 1
1
M.(M + KS)+
2 2 P.(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 h 0 (M + KS) ≥ 2 and by Lemma 3.6, (P + Q)C0 contributes to the Clifford index
of C0,and
Cliff (P + Q)C ≤ (P + Q) . (M + KS)
= Q.(M + P + KS) − Q.P + P.(M + KS)
= Cliff QC +2δ + M.P− Q.P + P.KS .

Math. Nachr. 256 (2003) 71
By (3.33) and (3.34), we get
�
Cliff (P + Q)C − 2+Q.P − P.KS , if Q �∼ −KS8 ,
Cliff A ≥
Cliff (P + Q)C − 3 − 2P .KS8 , if Q ∼ −KS8 .
(3.37)
If Q �∼ −KS8, (3.37) gives that Q.P − P.KS ≤ 2, which means that either −P .KS =2and Q.P =0,
or −P .KS =1and Q.P ≤ 1. In the first case we get P.(M + Q + KS) ≤ 2+0− 2=0, whence all the
smooth <strong>curves</strong> in |L| would be hyperelliptic by Lemma 3.6, a contradiction on the exceptionality of C0. Inthe
second case, the Hodge index theorem would give the contradiction Q ∼ P ∼−KS8.
If Q ∼−KS8, (3.37) gives that −P .KS8 =1, whence P ∼−KS8 (by (P6) and the fact that P ≥−KS).
Since M.P = −M .KS8 ≤ 2, the Hodge index theorem together with the assumptions M.(M + KS) ≥ 2,
gives M ∼−2KS8. Hence L ∼−4KS8. But this case belongs to the special cases of Serrano described in
Example 3.8 above, where we saw that none of the smooth <strong>curves</strong> in |L| were exceptional, a contradiction.
This proves that M.(M + KS) ≤ 0, and by symmetry we obtain the same result for P .
This shows (3.36).
Case II–a: Q �∼ −KS8 and P.M≤ 1.
We first show that both P 2 > 0 and M 2 > 0. Assume to get a contradiction that P 2 ≤ 0. ThenP.(P +KS) ≤
−2 and by (3.35) and (3.36), combined with h 0 (M + P + KS) ≥ 2,wegetM.P+ δ ≥ 2, and by (3.33) we get
the contradiction
Cliff A ≥ Cliff QC0 − P.M+2 ≥ Cliff QC0 +1.
The same argument works for M,sowehaveP 2 > 0 and M 2 > 0.
We then see by (3.33), (3.35) and (3.36), combined with h 0 (M + P + KS) ≥ 2, the Hodge index theorem
and (P6) that δ =0and M ∼ P ∼−KS8. Now (3.32) and (3.33) again give
Cliff A ≥ Cliff QC − 1 = −KS8 .Q− 1 . (3.38)
ButbyLemma3.6wehave
Cliff � �
− KS8
C0
≤ −KS8 .Q,
and since C0 is exceptional we must have equality in (3.38), whence also in (3.33), with
τ = τ ′ = ∅ , (3.39)
and moreover, by the remark right after (3.34), we have
c2(R1) − 2rk R1 = −Q.KS , if Q 2 > 0 . (3.40)
By (2.10), (3.6), (3.13) and (3.39), we get
h 0� � 0
R1 ⊗ KS8 = h � � 1
E ⊗ KS8 − 2 = h (A) − 2 ≥ 2 . (3.41)
By Corollary 3.3 we therefore have c2(R1) − 2rk R1 ≥−2, which by (3.40) means that Q.KS = −2 if Q 2 > 0.
This last equality also holds if Q 2 =0, whence by (3.38) we get Cliff A =1,soC0 is a smooth plane quintic
with h 0 (A) =3, a contradiction.
Case II–b: Q �∼ −KS8 and P.M=2.
By (3.33), we have δ =0and Cliff A = Cliff QC0, and by our assumptions that A computes the Clifford
dimension of C0 we have, by using (3.35) and (3.14)
whence
4 ≤ h 0 (A) ≤ h 1� QC0
� 0
= h (M + P + KS) = 1
1
M.(M + KS)+ P.(P + KS)+3,
2 2
M.(M + KS)+P.(P + KS) ≥ 2 , (3.42)

72 Knutsen: <strong>Exceptional</strong> <strong>curves</strong> on Del Pezzo surfaces
which is incompatible with (3.36).
Case II–c: Q ∼−KS8 and P.M≤ 1.
We first show that both P 2 > 0 and M 2 > 0. Assume to get a contradiction that P 2 ≤ 0. ThenP.(P +KS) ≤
−2 and by (3.35) and (3.36), combined with h 0 (M + P + KS) ≥ 2,wegetM.P+ δ ≥ 2, and by (3.34) we get
Cliff A ≥ Cliff QC0 − P.M+1,
and this gives us the only possibility P.M = δ =1. But then Cliff A = Cliff QC0 and by (3.15) we get
h0� �
0
QC0 =3 0 and M 2 > 0.
By the Hodge index theorem we therefore must have P ∼ M and M 2 = 1. If M �∼ −KS8, wehave
M.(M + KS) ≤−2 and by (3.35), again combined with h 0 (M + P + KS) ≥ 2,wegetδ ≥ 2. But then (3.34)
yields the contradiction
Cliff A ≥ Cliff QC0 +2.
Therefore we have P ∼ M ∼ Q ∼−KS8 and L ∼−3KS8, again a case belonging to Example 3.8 above, a
contradiction.
Case II–d: Q ∼−KS8 and P.M=2.
By (3.34) and the fact that C0 is assumed to be exceptional, we have δ =0and
Cliff A = Cliff QC0 − 1 = −KS8. � �
L +2KS8 − 1 = −L.KS8 − 3 , (3.43)
where the middle equality follows from Lemma 3.6 and the fact that δ =0.
By assumption h 1 (A) ≥ h 0 (A) ≥ 4, so by (2.1) and (3.43), we get
whence
8 ≤ g(L)+1− Cliff A = 1
2 L.� �
L + KS8 +2+L.KS8 +3 = 1
2 L.� �
L +3KS8 +5,
L. � �
L +3KS8 ≥ 6 . (3.44)
But then one easily computes
L. � � � � � �
L +3KS8 = M. M + KS8 + P. P + KS8 +2,
and therefore (3.44) is incompatible with (3.36).
This finally concludes the proof of Proposition 3.9.
As an immediate consequence we get
Corollary 3.10 Given the assumptions in Proposition 3.9. We have
DC0 ∼ A, h 0� � 0
DC0 = h (D) and Cliff C0 = D.(L + KS − D) .
Proof. Since h 0 (R ⊗ ωS) = 0 by the last proposition, we get by Proposition 3.5 (c) and tensoring the
sequence (3.6) with ωS that
h 1� DC0
� = h 0 (M + KS) = h 0 (E ⊗ ωS) (2.10)
= h 1 (A) ≥ 3 ,
whence DC0 contributes to the Clifford index of C0 and we must then have Cliff A = Cliff DC0 by Proposition
3.5 (d). Since we just saw that h1� �
1 0
DC0 = h (A) it follows from (2.1) that h � �
0
DC0 = h (A) also. Since
DC0 ≥ A and DC0 is base point free (because it computes the Clifford index of C0), we must have DC0 ∼ A.
Finally, since Cliff A = Cliff DC0, one easily sees that we must have equality in (3.10) and (3.11), whence
h1 (−M) =0, h0� �
0
DC0 = h (D) andbyLemma3.6Cliff DC0 = D.(L + KS − D).

Math. Nachr. 256 (2003) 73
Set r := h0 (A) − 1, which is by assumption the Clifford dimension of C0. We have then, by the last corollary,
Riemann–Roch and the adjunction formula:
r = 1
2 D.(D − KS) = D 2 +1− g(D) = g(D) − 1 − D.KS . (3.45)
By [8, Prop. 3.2] any pencil of divisors on C0 has degree ≥ 2r. In particular, since h0�OC0(−KS) � ≥
h0 (−KS) ≥ 2,wemusthave−KS .L≥ 2r =2g(D) − 2 − 2D.KS, which yields
−M .KS ≥ −D.KS +2g(D) − 2 . (3.46)
Now if g(D) ≥ 2,wehaveh 0 (D + KS) ≥ 2,sobyLemma3.6,OC0(D + KS) contributes to the Clifford index
of C0. Combining with (3.46), we get the contradiction
Hence
Cliff OC0(D + KS) ≤ (D + KS) .M = D.M + M.KS < D.M + D.KS = Cliff C0 .
g(D) ≤ 1 . (3.47)
By (3.45), this yields
�
D
r =
2 +1 if g(D) = 0,
D2 if g(D) = 1.
(3.48)
Now define d := D.L and c := Cliff C0 = D.(L + KS − D). By [8, Lemma 1.1], the line bundle
A = DC0 is very ample, whence it embeds C0 as a smooth curve C ′ 0 of degree d and genus g in Pr . Moreover,
since H0� �
0
DC0 � H (D), we have the following commutative diagram, where we denote by φD and φDC the 0
morphisms defined by D and DC0 respectively and set S ′ := φD(S):
S��
C0
φD ��S ′ �� r
�� P
φDC0 �� C ′ 0
�� P r
(3.49)
We therefore have deg S ′ ≤ D 2 . By [8, Cor. 3.5 and Prop. 5.1], a curve of Clifford dimension r ≥ 4 in P r of
degree d and genus g with (d, g) �= (4r − 3, 4r − 2) cannot be contained in a (possibly singular) surface of degree
≤ 2r − 3. By (3.48) we get
deg S ′ ≤ D 2 ≤ r ≤ 2r − 3
(when r ≥ 4). So we therefore must have
(d, g) = (4r − 3, 4r − 2) if r ≥ 4 . (3.50)
Now we make use of the following result:
Proposition 3.11 [8] Let C be a smooth curve of genus g, Clifford index c, Clifford dimension r ≥ 3 and
degree d in Pr . Consider the following properties:
(a) (d, g) =(4r−3, 4r − 2) (whence c =2r− 3),
(b) C has a unique line bundle A computing c,
(c) A2 � ωC and A embeds C as an arithmetically Cohen–Macaulay curve in Pr ,
(d) C is 2r–gonal, and there is a one–dimensional family of pencils of degree 2r, all of the form |A − B|,
where B is the divisor of 2r − 3 points of C.
If r ≤ 9,thenCsatisfies (a) – (d).
In general, if C satisfies (a), it also satisfies (b) – (d).

74 Knutsen: <strong>Exceptional</strong> <strong>curves</strong> on Del Pezzo surfaces
Remark 3.12 In [8] it is furthermore conjectured that any curve of Clifford dimension r ≥ 3 satisfies (a) – (d).
By this result and (3.50) we have
Lemma 3.13 C0 satisfies the conditions (a) – (d) in Proposition 3.11 if r ≥ 3.
By condition (c) we have OC0(2D) � ωC0 �OC0(L + KS), so we have an exact sequence
0 −→ KS − 2D −→ L + KS − 2D −→ OC0 −→ 0 ,
from which we get h 0 (L + KS − 2D) =1. � Note that h 1 (KS − 2D) =h 1 (2D) =0by (P5). � This means that
L + KS − 2D =: ∆ ≥ 0 , (3.51)
where ∆ .L=0and ∆ is only supported on (−1)–<strong>curves</strong>.
Furthermore, from Lemma 3.6 and the fact that D is nef (Lemma 3.4 (b)), we must have ∆ .D =0, whence
by (3.51) ∆ 2 =∆.KS. Riemann–Roch therefore yields
From the exact sequence
we therefore get
h 1 (∆) = h 0 (∆) − 1
2 ∆ . (∆ − KS) − 1 = 0.
0 −→ KS − ∆ −→ 2D −→ OC0(2D) −→ 0 ,
h 0 (2A) = h 0� OC0(2D) � = h 0 (2D) = D.(2D − KS)+1 = 2r + D 2 +1. (3.52)
By Lemma 3.13 and condition (c) in Proposition 3.11, we have h 0 (2A) =4r − 2 if r ≥ 3. Combining (3.48)
and (3.52) yields
4r − 2 = h 0� OC0(2D) � = 2r + D 2 +1 = 3r + g(D) ,
whence r = D 2 =3and g(D) =1.
We have therefore proved that an exceptional curve on a Del Pezzo surface has Clifford dimension 2 or 3.
3.1 Curves of Clifford dimension 2
From (3.48), we see that D2 =2if g(D) =1. Hence c = D.L− D2 + D.KS = d − 4. In particular d ≥ 5.
On the other hand, since D satisfies the conditions (†) for k = d − 3 ≥ 2, we get from Lemma 3.7 above that
all � the smooth <strong>curves</strong> in |L| have gonality ≤ d − 2, which contradicts the exceptionality of C0. Sowemusthave
2 r, g(D),D � =(2, 0, 1).
In particular D.KS = −3, so by [11, Prop. 3.10] we have that φD is an isomorphism outside of finitely many
contracted (−1)–<strong>curves</strong>. In particular it is birational.
By the commutative diagram (3.49) with r =2,wehavethatφD : S → P2 is a blow up at finitely many
points, which is equal to the number m(D). Clearly 0 ≤ m(D) ≤ 8 and S � Sm(D). Let
R(D) = {Γ1,...,Γ m(D)} . (3.53)
Since DC0 is very ample, we must have
m(D) �
C0 ∼ dD − aiΓi , (3.54)
i=1
with all ai =0or 1. InotherwordsΓi .L =0or 1 for all i. ThismeansthatDC is very ample for any smooth
C ∈|L|, so any such is isomorphic to a smooth plane curve of degree d and is exceptional.

Math. Nachr. 256 (2003) 75
3.2 Curves of Clifford dimension 3
From (3.48), we have D2 =3and g(D) =1, whence −D.KS =3. By the Hodge index theorem K2 S ≤ 3
with equality if and only if D ∼−KS6. It is known from the results of G. Martens in [15] (and also contained in
Proposition 3.11 (a)) that a curve C0 of Clifford dimension 3 satisfies
g(C0) = 10, c := Cliff C0 = 3, d := D.L = 9 and (2D)C0 ∼ ωC0 . (3.55)
The Hodge index theorem gives 3L 2 = D 2 L 2 ≤ (D.L) 2 =81, whence L 2 ≤ 27 with equality if and only
if L ∼ 3D. Since g(L) =10,wehaveL.(L + KS) =18, and combined with the Hodge index theorem
K 2 S L2 ≤ (L.KS) 2 = � L 2 − 18 � 2 , we end up with the following possibilities:
(i) L 2 =23, −L.KS =5, K 2 S =1.
(ii) L 2 =24, −L.KS =6, K 2 S =1.
(iii) L2 =25, −L.KS =7, K2 S =1.
(iv) L2 =26, −L.KS =8, K2 S
=1, 2.
(v) L2 =27, −L.KS =9, K2 S =1, 2, 3 � K2 �
S =3implying L ∼ 3D ∼−3KS6 .
We now show that we can rule out the two first cases.
Indeed, in case (i), the pencil � �
� − KS8
� will cut out a g1 5 on each member of |L|, in particular on C0, contradicting
its exceptionality.
In case (ii) we calculate
χ(L − 2D) = 1 � � 2 2
L − 4L.D+4D − KS8 .L+2D.KS8 +1 = 1,
2
whence either L − 2D ≥ 0 or KS8 − L +2D ≥ 0. But � KS8 − L +2D � .L= −12, and since L is nef, we must
have L − 2D ≥ 0. Since(L−2D) .KS8 =0,wemusthaveL∼2D, but this contradicts (L − 2D) .L=6.
We now consider the cases (iii) – (v).
First note that D satisfies the conditions (†) for k =5. We claim that there is no line bundle satisfying (†) for
k

76 Knutsen: <strong>Exceptional</strong> <strong>curves</strong> on Del Pezzo surfaces
In the same way we can treat the case (v) and we leave it to the reader to verify that we end up with the
following cases:
L ∼ −3KS8 +3Γ1 +3Γ2 , R(L) = {Γ1, Γ2} , Γ1 . Γ2 = 0, (3.60)
L ∼ −3KS7 +3Γ, R(L) = {Γ} , (3.61)
L ∼ −3KS6 . (3.62)
We now claim that any smooth C ∈|L| for an L as in (3.57) – (3.62) is exceptional. Indeed, note that any
smooth curve in |L| for L as in (3.57) – (3.61) is a strict transform of some smooth curve in |−3KS6|, fora
blowingupofS6 in one or two points (in general position), so what we claim now follows from Example 3.1
above.
We have therefore proved:
Theorem 3.14 Let C0 be a smooth curve of Clifford index c and Clifford dimension r ≥ 1 on a Del Pezzo
surface S. LetAbe a line bundle computing its Clifford dimension. Set L := OS(C0).
Then C0 is exceptional if and only if it is as in one of the following two cases:
(a) There exists a base point free line bundle D on S satisfying
D 2 = 1, D.KS = −3 , D.L = c +4 and
Γi .L ≤ 1 for all Γi ∈R(D)
(3.63)
(the latter condition is equivalent to DC being very ample for all C ∈|L|).
In this case r =2, A ∼OC0(D) and φD : S → P 2 gives the identification S � S m(D) and maps every
smooth curve in |L| to a smooth plane curve of degree D.L= c +4, whence all these <strong>curves</strong> are exceptional of
Clifford dimension 2 and Clifford index c.
(b) S � Sn for n ∈{6, 7, 8} and
n−6 �
L ∼ −3KSn +
i=1
aiΓi , Γi = (−1)–curve , ai ∈{2, 3} .
In this case r =3, g(L) =10, c =3and A ∼OC0(D), for
n−6 �
D := −KSn +
i=1
Γi
and all the smooth <strong>curves</strong> in |L| are exceptional of Clifford dimension 3 and Clifford index 3.
This concludes the proof of part (c) of Theorem 1.1.
It is clear that after a change of basis of Pic S we can assume that we have
L ∼ dl −
n�
aiei , ai∈{0, 1} , and D ∼ l,
i=1
in case (a), and that we have
in case (b).
L ∼ 9l −
6�
ei −
i=1
n�
biei , bi∈{0, 1} , and D ∼ 3l −
i=7
4 Conclusion of the proof of Theorem 1.1
From [11] we have the following result:
6�
i=1
ei
(3.64)

Math. Nachr. 256 (2003) 77
Proposition 4.1 [11, Prop. 4.1] Let L be a nef line bundle of sectional genus g(L) ≥ 2 on a Del Pezzo surface
such that L �∼ −2KS8. Assume that the minimal gonality of a smooth curve in |L| is k +1, for an integer k ≥ 1
and that among all the divisors satisfying (†), there is one not being of one of the following types:
(a) k =1, D ∼−KS7 and L ∼−2KS7,
(b) k =1, D ∼−KS8 +Γ, with Γ a (−1)–curve and L ∼ 2D,
(c) D ∼−KS8.
Then all the smooth <strong>curves</strong> in |L| have gonality k +1.
In [11, Section 4] we furthermore showed that the cases (a) and (b) are exactly the cases (I) and (II) in
the introduction. We also showed that if the only divisor satisfying (†) for k = k0 is −KS8, then there is a
codimension 1 family of smooth (k0 +1)–gonal <strong>curves</strong> in |L|, consisting precisely of the <strong>curves</strong> passing through
the base point x of � �
� − KS8
�, whereas all the other <strong>curves</strong> are (k0 +2)–gonal.
We have therefore left to prove that the case (c) above is Case (III) in the introduction. We have to prove the
numerical conditions, which follow by:
Lemma 4.2 Let L be a base point free line bundle of sectional genus g(L) ≥ 2 on a Del Pezzo surface of
degree 1 and k0 ≥ 1 an integer. Assume that the minimal gonality of a smooth curve in |L| is k0 +1. The
following conditions are equivalent:
(i) the only divisor satisfying (†) for k = k0 is −KS,
(ii) L2 ≥ 5k0 +2, −L.KS = k0 +2, L is ample, L.C ≥ k0 +2for all smooth rational <strong>curves</strong> C with
C2 =0, and L.Γ ≥ 2 for all (−1)–<strong>curves</strong> Γ if L2 ≥ 8.
P r o o f. Assume (i) holds. As remarked right after Proposition 4.1 above, we have that the general smooth
curve in |L| has gonality k0 +2.Sincek0 +2 ≤⌊ g(L)+3
2 ⌋ (see Section 2.4), we get g(L) ≥ 2k0 +1.Furthermore
−KS .L= k0 +2and L2 =2g(L) − 2 − L.KS ≥ 5k0 +2.
By [11, Prop. 3.7] we have that L + KS is nef, whence R(L) =∅. If there exists a (−1)–curve Γ such
that Γ .L =1,defineD := −KS +Γand M := L − D = L + KS − Γ. If L2 ≥ 8, one easily computes
h0 (L − 2D) ≥ 1
� �
2 2
2 L − 3k − 6 +1> 0 and L.D− D = k0 +1,soD will satisfy the conditions (†) for
k = k0, a contradiction.
If there is a smooth rational curve C with C2 =0and L.C ≤ k0 +1,thenCsatisfies (†) for k = k0,againa
contradiction.
Assume now (ii) holds and that D is a divisor satisfying (†) for k = k0. By [11, Prop. 3.7 and Rem. 3.8] either
g(D) =0and � D2 �
, −D.KS =(0, 2) or (1, 3),org(D) =1and D is of the form (3.56).
If L 2 =7(so that k0 =1),wefind(−2KS +Γ) 2 =(−2KS +Γ).L =7, whence by the Hodge index
theorem L ∼−2KS +Γ, and the only divisor satisfying (†) for k = k0 is −KS.
So we can assume L2 ≥ 8, whence by our assumptions R(L) =R1(L) =∅,soifg(D) =1,thenD∼−KS by (3.56). So we have left to treat the cases where � D2 �
, −D.KS =(0, 2) or (1, 3).
By the assumptions in (ii) one immediately sees that D cannot be as in the first case.
So assume now D2 =1and D.KS = −3. Sinceg(L) ≥ 2k0 +1,wehave(k0,g(L)) �= (2, 3), whence
by [11, Prop. 3.7] all the smooth <strong>curves</strong> in |L| are exceptional, and by Theorem 3.14 (a) Γi .L ≤ 1 for any
Γi ∈R(D). (Note that R(D) �= ∅, in fact m(D) =8, by Theorem 3.14 (a).) This contradicts the conditions
(ii).
This concludes the proof of parts (a) and (b) of Theorem 1.1.
We have shown above that if L is as in Case (III) with L 2 =7,thenL ∼−2KS +Γfor a (−1)–curve Γ.
Conversely any such L is easily seen to satisfy the conditions of Case (III).
For L 2 ≥ 8, a line bundle in Case (III) is determined by its intersection numbers with <strong>curves</strong> D on S with
D 2 = −1 and D 2 =0. Now Pic S8 � Zl ⊕ Ze1 ⊕ ... ⊕ Ze8 and −KS8 =3l − � 8
1 ei as described in
Section 2.3. Writing D ∼ bl − � 8
1 biei with b1 ≥ ... ≥ b8 we get finitely many tuples (b; b1,...,b8) yielding
D 2 = −1 and D 2 =0(see e.g. [5, pp. 4 – 5]). More precisely, writing � a0; a n1
1 ,an2
2 ,...� for the class of the
curve a0l − �n1 1 eit − �n2 1 ejt − ...we get the following seven types of <strong>curves</strong> D with D2 = −1:
� �
0; −1 ,
� 2
1; 1 � ,
� 5
2; 1 � ,
� 6
3; 2, 1 � ,
� 3 5
4; 2 , 1 � ,
� 6 2
5; 2 , 1 � ,
� 7
6; 3, 2 � ,

78 Knutsen: <strong>Exceptional</strong> <strong>curves</strong> on Del Pezzo surfaces
and the following fifteen types of <strong>curves</strong> D with D2 =0:
� � � 4
1; 1 , 2; 1 � , � 3; 2, 1 5� , � 4; 3, 1 7� , � 4; 2 3 , 1 4� , � 5; 2 6 , 1 � , � 5; 3, 2 3 , 1 4� , � 6; 3 2 , 2 4 , 1 2�
� � � 4 3 6
7; 3 , 2 , 1 , 7; 4, 3, 2 � , � 8; 3 7 , 1 � , � 8; 4, 3 4 , 2 3� , � 9; 4 2 , 3 5 , 2 � , � 10; 4 4 , 3 4� , � 11; 4 7 , 3 � .
Now we set L ∼ al − � 8
1 aiei, and we can assume a1 ≥ ...≥ a8 by symmetry. The condition −L.KS = d
now reads d =3a − a1 − ...− a8 and L 2 ≥ 5d − 8 gives
a 2 ≥ a 2 1 + ...+ a 2 8 +5d − 8 = a 2 1 + ...+ a 2 8 +15a − 5a1 − ...− 5a8 − 8 .
Posing the remaining conditions on L we leave it to the reader to verify that we get the following alternative
description of the line bundles in Case (III):
Alternative description of Case (III): deg S =1, and either L2 =7,g(L) =3,d=3,L∼−2KS +Γ
for a (−1)–curve Γ, orL ∼ al − �8 1 aiei, d := 3a − a1 − ...− a8, with the following conditions, assuming
a1 ≥ ...≥ a8:
a8 ≥ 2 , a7 + a8 ≥ a1 , a5 + ...+ a8 ≥ a, a2 + ...+ a8 ≥ 2a,
a ≥ a1 + a2 +2, a ≥ 2a1 , a+ a8 ≥ a1 + a2 + a3 ,
2a ≥ a1 + ...+ a5 +2, 2a ≥ 2a1 + a2 + a3 + a4 , 2a + a8 ≥ a1 + ...+ a6 ,
3a ≥ 2a1 + a2 + ...+ a7 +2, 3a ≥ 2a1 +2a2 + a3 + ...+ a6 ,
4a ≥ 2a1 +2a2 +2a3 + a4 + ...+ a8 +2,
4a ≥ 3a1 +2a2 + a3 + ...+ a8 ,
4a ≥ 2a1 + ...+2a4 + a5 + a6 + a7 ,
5a ≥ 2a1 + ...+2a6 + a7 + a8 +2, 5a ≥ 2a1 + ...+2a7 ,
5a ≥ 3a1 +2a2 + ...+2a5 + a6 + ...+ a8 ,
6a ≥ 3a1 +2a2 + ...+2a8 +2, 6a ≥ 3a1 +3a2 +2a3 + ...+2a7 + a8 ,
7a ≥ 3a1 + ...+3a4 +2a5 + ...+2a8 ,
8a ≥ 3a1 + ...+3a7 +2a8 ,
a(a − 15) ≥ a1(a1 − 5) + ...+ a8(a8 − 5) − 8 .
5 Pencils computing the gonality
In Theorem 3.14 we described the line bundles A computing the Clifford dimension of smooth exceptional <strong>curves</strong>
on a Del Pezzo surface. In particular, we saw that such a line bundle is the restriction of a line bundle on the
surface.
We conclude this paper by describing the pencils computing the gonality of a smooth curve on a Del Pezzo
surface. This will make some of the results of Pareschi more precise (see [17, Lemma (2.8)]), and extend them
to Del Pezzo surfaces of degree 1.
By Lemma 3.7, if A computes the gonality of a smooth curve C on a Del Pezzo surface S, then there exists
a line bundle D on S such that OC(D) ≥ A. Furthermore, if the gonality is the minimal among the smooth
<strong>curves</strong> in |C| (which is always the case, except possibly for the cases (I), (II) and (III) described above), the line
bundle D has to be as in (a) – (f ) of [11, Prop. 3.7], i.e. either D ∼−KS8 or � D2 �
, −D.KS =(0, 2), (1, 3),
(2, 2), (3, 3) or (4, 4), the latter implying that the gonality is 4. Furthermore, by [11, Prop. 3.7] and Theorem
3.14 combined, if � D2 �
, −D.KS =(3, 3),then(gon C, g(C)) = (3, 4), (4, 5) or (6, 10).
In the special cases (I), (II) and (III), if C has minimal gonality d − 1, then we have seen that D ∼−KS7,
D ∼−KS8 +Γ(where Γ is a (−1)–curve) and D ∼−KS8 respectively, and there are no other line bundles
satisfying (†) for k = d − 2 by Proposition 4.1. If C has gonality d, a careful analysis is needed. Denoting by
A again a line bundle computing the gonality, by Lemma 3.7 there exists an effective divisor D ′ on S satisfying
(†) for k = d − 1 � in particular D ′ .L− D ′2 �
≤ d such that OC(D ′ ) ≥ A. If D ′ �∼ D, then we must have
D ′ .L− D ′2 ′ ′ ′ ′ = d. IfD is not nef, then there is a (−1)–curve Γ satisfying Γ .D < 0. By arguing as in [11,
(3.9)], we find that Γ ′ .D ′ =1, Γ ′ .L =0and D ′ − Γ ∼ D. Since L is ample in the cases (I) and (III), we

Math. Nachr. 256 (2003) 79
must be in case (II), and we also see that Γ ′ =Γ, whence D ′ ∼−KS8 +2Γ. But since Γ .L =0,wehave
OC(D ′ ) �OC(D), whence we can work with D ∼−KS8 +Γin this case. So we can assume that D ′ is nef. By
the same argument, we also get that M ′ := L − D ′ is nef, except when we are in case (II) and M ′ ∼−KS8 +2Γ.
In this case D ′ ∼−KS8, and again since Γ .L =0,wehaveOC(D ′ ) �OC(D), whence we can assume that
M ′ is nef as well.
We now treat the three different special cases and will find out that such a divisor D ′ does not exist in the cases
(I) and (II), and that there are a few possibilities in case (III).
Case (I): We have d =3and L ∼−2KS7. In particular −2KS7 .D ′ = D ′ .L = D ′2 +3.SinceL − D ′ ≥
−KS,wehaveD ′ .L≤ (L + KS) .L=4, whence L.D ′ =4, D ′2 =1and −KS7 .D ′ =2, a contradiction on
the parity.
Case (II): This case is similar to case (I) and left to the reader.
Case (III): We have deg S =1and −L.KS8 = d ≥ 3.
We first treat the case g(D ′ ) ≥ 2. Thenh 0 (D ′ + KS) ≥ 2.
Assume first −D ′ .KS ≤−M ′ .KS. Wehave
(D ′ + KS) . (M ′ − KS) = D ′ .M ′ + KS . (M ′ − D ′ ) − K 2 S < D′ .M ′ , (5.1)
whence D ′ + KS8 ∼−KS8, i.e.D ′ ∼−2KS8. One has d = D ′ .L− D ′2 = −2KS8 .L− 4K2 =2d− 4,
S8
whence d =4.Wehaveh0�D ′ �
′
C =4and deg D C =8, so there exists an effective divisor Z of degree 4 on
C such that A ∼ D ′ C − Z. This implies that h0�D ′ �
0 ⊗IZ = h (A) =2, so there are two distinct elements
D1 and D2 in |D ′ | passing through Z. Since the only reducible elements of |D ′ | are of the form H1 + H2 for
Hi ∈ � �
� �
� − � KS8 and C does not pass through the base point of � − � KS8 , one easily sees that D1 and D2 must be
reduced and irreducible, in particular they intersect properly. Since D1 .D2 =degZ =4,wehaveZ = D1∩D2.
Conversely, picking any two irreducible elements D1 and D2 in � �
� − � 2KS8 intersecting in 4 distinct points and
setting Z := D1 ∩ D2, one sees from the exact sequence
0 −→ M ′ −→ L ⊗IZ −→ M ′ D2
−→ 0
that |L ⊗IZ| is nonempty and has no base points off Z, whence from Bertini’s theorem the generic member is
smooth. For any such smooth C, the line bundle � �
− 2KS8 C − Z is then a g1 d .
Assume now −D ′ .KS > −M ′ .KS. If −M ′ .KS =1then M ′ ∼−KS8by (P6), whence the absurdity
d = D ′ .M ′ = −D ′ .KS8 = − � �
L + KS8 .KS8 = d − 1. So −M ′ .KS8 ≥ 2, which in particular yields
d = −L.KS8 ≥ 5.
Furthermore, by (5.1) with D ′ and M ′ interchanged, we have g(M ′ )=1, unless possibly if M ′ ∼−2KS8.
But in this case, we get d =4as above, a contradiction. So g(M ′ )=1.
By the conditions (†) and [11, Lemma 3.1 and Rem. 3.4 (ii)], we must have g(L) =2d − 3 or 2d − 2 and
L 2 ≤ 4d−2. Since−L.KS = d,wegetL 2 =2g(L)−2−L.KS =5d−6 (resp. 5d−8)ifg(L) =2d−2(resp.
g(L) =2d − 3). Sinced ≥ 5 we end up with the only possibilities (d, g(L)) = (5, 7) and (6, 9). In the first case,
we get g(D ′ )=2, whence −D ′ .KS =3and D ′2 =5. In particular D.L =10, whence A ∼ D ′ C − Z5, for
Z5 a scheme of length 5. Similarly, we get g(D ′ )=3and −D ′ .KS =4in the second case, whence D ′2 =8,
D ′ .L=14,andA∼ D ′ C − Z8,forZ8a scheme of length 8.
The cases that are left are now g(D ′ )=0or 1. By computing Cliff D ′ C as in the proof of [11, Prop. 3.7], we
leave it to the reader to check that we end up with only three possibilities:
(1) g(D ′ )=0, D ′2 =0,
(2) g(D ′ )=0, D ′2 =1, d =3,
(3) g(D ′ )=1, D ′2 =2.
Summarizing, we obtain the following complete classification:
Proposition 5.1 Let C be a smooth curve of genus g ≥ 2 on a Del Pezzo surface and A a line bundle
computing its gonality.
(a) If C is of gonality d and not as in the cases (I), (II) and (III),thenAisof one of the following types:
(i) A = DC,forD2 =0, −D.KS =2,
(ii) A = DC − P , P any point of C, D2 =1, −D.KS =3,
(iii) A =(−KS8)C − x, x base point of |−KS8|,

80 Knutsen: <strong>Exceptional</strong> <strong>curves</strong> on Del Pezzo surfaces
(iv) A = DC − Z2, D2 = −D.KS =2, Z2 a length 2 scheme where D fails to be very ample, K2 S ≤ 2,
(v) A = DC − Z3, D2 = −D.KS =3, Z3 a length 3 scheme where D fails to be 2–very ample, (d, g) =
(3, 4), (4, 5) or (6, 10), K2 S ≤ 3,
(vi) A = DC − Z4, D2 = −D.KS =4, Z4 a length 4 scheme where D fails to be 3–very ample, (d, g) =
(4, 5), K2 S ≤ 4.
(b) If C is as in case (I) (resp. (II)), thenA = � � � �
− KS7 − Z2 resp. A = − KS8 +Γ C �
C − Z2, with
Γ a (−1)–curve � �
, with Z2 a length 2 scheme where −KS7 resp. −KS8 +Γ � fails to be very ample if C is
hyperelliptic; and A = � �
− KS7 C − P � resp. A = � − KS8 +Γ �
C − P � for any point P ∈ C if C is trigonal.
(c) If C is as in case (III) and of minimal gonality d − 1,thenA = � �
− KS8 − x, wherexisthe base point
C
of |−KS8|. IfChas gonality d (equivalently C does not pass through x),thenAisof one of the following types:
(i) A = � �
− KS8 C ,
(ii) A = � �
− 2KS8 C − Z4, whereZ4has length 4 and Z4 = D1 ∩ D2 for two distinct <strong>curves</strong> D1 and D2
in |−2KS8|,
(iii) A = DC, forD2 =0, −D.KS =2,
(iv) A = DC − P , P any point of C, D2 =1, −D.KS =3, d =3,
(v) A = DC − Z2, D2 = −D.KS =2, Z2 a length 2 scheme where D fails to be very ample,
(vi) A = DC − Z5, D2 =5, −D.KS =3, Z5 a length 5 scheme, (d, g) =(5, 7),
(vii) A = DC − Z8, D2 =8, −D.KS =4, Z8 a length 8 scheme, (d, g) =(6, 9).
We also make the following observation:
Corollary 5.2 Let S be a Del Pezzo surface of degree n ≥ 3 embedded in Pn by −KS, and let C ⊆ S be a
smooth curve of genus g ≥ 6 which is not exceptional. Then any pencil computing the gonality of C is cut out
on C by a pencil of conics on S, whence the number of such pencils on C is finite. (In particular, S contains no
bi–elliptic curve of genus ≥ 6.)
Acknowledgements The author would like to thank S. A. Strømme for help in correcting a mistake in an earlier version of
the paper and a referee for useful comments, in particular for suggesting to include Cor. 5.2.
The author was supported by a grant from the Research Council of Norway. Most of the paper was written at the University
of Bergen, Norway.
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