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L’irrégularité des surfaces de type général dont le système canonique est composé d’un pinceau. (The irregularity of surfaces et general type whose canonical system is composed of a pencil). (English) Zbl 0594.14029
Let S be a surface of general type over \({\mathbb{C}}\). Suppose that the canonical system \(| K_ S|\) is composed of a pencil. Then up to birational equivalence, there is a fibration \(f: S\to B\) onto a smooth curve B of genus b such that the general fibre is a curve of genus \(\geq 2\). It turns out that such a surface is rather restrictive. Namely, the present paper proves that either \(q(S)=b=1\) or \(b=0,q(S)\leq 2,\) where \(q(S)=\dim H^ 1(S,{\mathcal O}_ S)\). The proof consists of two parts. In the first part it is shown that if \(b>0\), then \(q=b=1\), which considerably improves the earlier result: \(q\leq b+1\), given by O. Debarre [Bull. Soc. Math. Soc. Fr. 110, No.3, 319-346 (1982; Zbl 0543.14026)]. The proof is based on a result of T. Fujita [J. Math. Soc. Japan 30, 779-794 (1978; Zbl 0393.14006)], which tells the positivity in a certain sense of the sheaf \(f_*\omega_{S/B}.\)
In the second part it is first shown by using a result of Castelnuovo that the image of S of the Albanese mapping is a curve. Then a detailed analysis of the two fibrations gives the bound: \(q\leq 2\).
Reviewer: F.Sakai

MSC:
14J10 Families, moduli, classification: algebraic theory
14J25 Special surfaces
14C20 Divisors, linear systems, invertible sheaves
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References:
[1] A. Beauville : L’inégalité pg \? 2q - 4 pour les surfaces de type général. Appendice à O. Debarre: ”Inégalités numériques pour les surfaces de type général” . Bull. Soc. Math. France 110 (1982) 319-346. · Zbl 0543.14026
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