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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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