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Non-hyperelliptic fibrations of small genus and certain irregular canonical surfaces. (English) Zbl 0822.14009
A nonsingular complex algebraic surface $$S$$ is called canonical if the rational map defined by the linear system $$| K_ S |$$ is birational onto its image. Let $$B$$ be a nonsingular curve, and let $$f : S \to B$$ be a fibration with connected fibers. If $$S$$ is canonical, then the general fiber of $$f$$ is a non-hyperelliptic curve. Let $$\omega_{S/B}$$ be the relative canonical sheaf. It is easy to see that the sheaf $$f_ * \omega_{S/B}$$ is a locally free sheaf of rank $$g$$ and degree $$\Delta (f) = \chi ({\mathcal O}_ S) - (g - 1) (b - 1)$$, where $$b$$ is the genus of $$B$$. If $$\Delta (f) \neq 0$$ (that is, the fibration is not locally trivial), then the number $$\lambda (f) = K^ 2_{S/B}/ \Delta (f)$$ is called the slope of $$f$$. Thus, to bound $$g$$ is is natural to bound the slope $$\lambda (f)$$. It is known that $$\lambda \geq 4 -{4 \over g}$$, and this bound is sharp for hyperelliptic fibrations.
In the paper under review the author proves that in the nonhyperelliptic case $$\lambda \geq {24 \over 7}$$ for $$g = 4$$ and $$\lambda \geq {40 \over 11}$$ for $$g = 5$$ (the previously known bound $$\lambda \geq 3$$ for $$g = 3$$ is also derived). In the course of proof, the author obtains information about the boundary cases. More generally, the author proves the conjecture of G. Xiao [Math. Ann. 276, 449-466 (1987; Zbl 0604.14024)] to the effect that for nontrivial nonhyperelliptic fibrations one always has $$\lambda > 4 -{4 \over g}$$. As an application, it is shown that the canonical image of an irregular canonical surface $$S$$ with a nonlinear pencil (i.e. with $$b > 0)$$ cannot be cut out by quadrics provided that $$K^ 2_ S \leq {10 \over 3} \chi ({\mathcal O}_ S)$$.
Reviewer: F.L.Zak (Moskva)

##### MSC:
 14C99 Cycles and subschemes 14C20 Divisors, linear systems, invertible sheaves 14J10 Families, moduli, classification: algebraic theory
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