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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
##### Keywords:
irregularity; surface of general type; canonical system
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##### References:
  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  E. Bombieri : Canonical models of surfaces of general type , Publ. Math. IHES 42 (1973) 171-219. · Zbl 0259.14005  G. Xiao : Finitude de l’application bicanonique des surfaces de type général , à paraître. · Zbl 0611.14031  A. Beauville : L’application canonique pour les surfaces de type général , Inv. Math. 55 (1979) 121-140. · Zbl 0403.14006  T. Fujita : On Kähler fiber spaces over curves , J. Math. Soc. Japan 30 (1978) 779-794. · Zbl 0393.14006  A. Beauville : Surfaces Algébriques Complexes , Astérisque 54, SMF (1978). · Zbl 0394.14014  U. Persson : Double coverings and surfaces of general type . In: Algebraic Geometry , Springer Lecture Notes in Math. 687 (1978). · Zbl 0396.14003
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