LINEAR SYSTEMS
ON WEAK DEL PEZZO SURFACES
E. BALLICO, C. FONTANARI and L. TASIN
We investigate the gonality and the Clifford index of smooth curves on weak Del Pezzo surfaces by following the lines of [12].
AMS 2010 Subject Classification: 14H51.
Key words: linear system, gonality, Clifford index, weak Del Pezzo surface.
1. INTRODUCTION
LetS be a smooth complex projective surface. The main question moti- vating the present note is the following: given a line bundle L on S, when is the gonality of the smooth curves in the linear system |L|constant?
Indeed, it is an elementary consequence of Noether’s theorem (see [1] or [7]) that smooth plane curves of the same degree have the same gonality. In the case of Hirzebruch surfaces, the gonality is computed by a ruling, which one obvious exception (see [10]). On the other hand, a counterexample to the constancy of gonality in a linear system on an elliptic ruled surface was exhibited in [8]. A celebrated counterexample in the case of K3 surfaces had been found by Donagi and Morrison [4], which turned out to be the only one according to Ciliberto and Pareschi [2]. Pareschi himself [12] had also shown that on a Del Pezzo surface of degree ≥ 2 the gonality of smooth curves is constant in a linear system with only one exception. More recently, Knutsen [9] generalized this result for Del Pezzo surfaces of degree 1, by classifying all exceptions.
Another natural question is about the constancy of the Clifford index in a linear system of curves on a surface. An affirmative answer was provided by [6] for K3 surfaces and by [12], Theorem B, for Del Pezzo surfaces.
Here instead we address the case in whichSis a weak Del Pezzo surface, namely a smooth projective surface with nef and big anticanonical divisor, and by generalizing Pareschi’s approach in [12] we obtain the following results.
REV. ROUMAINE MATH. PURES APPL.,55(2010),2, 93–96
94 E. Ballico, C. Fontanari and L. Tasin 2
Theorem 1. Let S be a weak Del Pezzo surface of degree ≥ 2 and let C ⊂S be a smooth curve of genus ≥2 such that C /∈ | −2ωS|. Then
gon(C0) = gon(C)
for every smooth curve C0 ∈ |C|. On the other hand, a general C ∈ | −2ωS| is trigonal, but the same systems contains smooth hyperelliptic curves as well.
Theorem 2. Let S be a weak Del Pezzo surface of degree ≥ 2 and let C ⊂S be a smooth curve of genus ≥4. Then
Cliff(C0) = Cliff(C) for every smooth curve C0 ∈ |C|.
2. THE PROOF
LetSbe smooth projective surface with nef and big anticanonical divisor (a so-called weak Del Pezzo surface). The degree ofSis defined as the positive integer ω2S.
The following properties are well-known (see [3], Corollary 3):
if deg(S)≥2 then ωS∨ is base point free;
(1)
h1(ωSn) = 0 for everyn∈Z; (2)
h0(ωSn) = (n(n−1)/2) deg(S) + 1 forn≤0 andh0(ωSn) = 0 for n >0.
(3)
From the exact sequence
0→ωS → OS(C)⊗ωS →ωC →0 we get immediately
(4) if C is a smooth curve on S withg(C)≥2 then h0(OS(−C)⊗ω∨S) = 0.
From the regularity ofS and from the exact sequence 0→ OS(−C)→ OS → OC →0 we have
(5) ifC is a reduced curve onS thenh1(OS(−C)) = 0.
Moreover,
(6) ifL is a base point free line bundle onS thenh1(L) =h2(L) = 0.
This follows from the fact that if L is base point free then it is nef and so L⊗ωS∨ is big and nef, hence we can apply Kawamata-Viehweg vanishing to h1(L) =h1((L⊗ωS∨)∨) and toh2(L) =h2((L⊗ωS∨)∨).
From the Hodge index theorem we deduce
(7) if deg(S)≥2, Lis base point free and L2 >0 then L·ωS∨ ≥2.
3 Linear systems on weak Del Pezzo Surfaces 95
We will need the following auxiliary result.
Lemma1. Let Sbe a Del Pezzo surface of degree2and letCbe a smooth curve on S such that g(C)≥2 and C /∈ | −2ωS|. If h0(ω∨S|C) = 3 then ωS∨|C is not very ample.
Proof. Suppose that ωS∨|C is very ample. Then it gives an immersion of C inP2 as a smooth plane curve of degree dand we have
−C·ωS=d.
From the genus formula
d(d−3) = 2g−2 =C2+C·ωS
we obtain
C2=d(d−2).
But then the Hodge index theorem gives
d2= (−C·ωS)2 ≥C2ωS2 = 2d(d−2).
The only admissible case is the equality with d = 4, which occurs for C ∈
| −2ωS|.
Proof of Theorem 1. The proof of Theorem 2.3 of [12] applies verbatim to our situation thanks to the above remarks and Lemma 1 and yields our claim forC /∈ | −2ωS|. If insteadC ∈ | −2ωS|, then we fall in the case of [12], Example (2.1). Indeed, the anticanonical morphism of S is generically finite and we can apply Riemann-Hurwitz ramification formula by [11], Theorem 2-1-9, hence the same conclusion follows.
Proof of Theorem 2. Let C be an smooth curve with minimal Clifford index among all smooth curves in its linear system. Let A be a line bundle computing the Clifford index of C. As in [12], proof of Theorem (3.1), we may assumeh0(C, A)≥3: indeed, ifA is a pencil then the by the analogue of [12], Lemma (2.8) and Example (2.7), C has genus at most 5, contradicting [5], pp. 174–175. Hence the rest of the proof of [12], Theorem (3.1), can be applied verbatim to our case thanks to the above remarks.
Acknowledgements. The authors are partially supported by MIUR and GNSAGA of INdAM (Italy).
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Received 12 January 2010 University of Trento
Department of Mathematics Via Sommarive 14 38123 Povo (TN), Italy {ballico,fontanar}@science.unitn.it
luca.tasin@unitn.it
