Irregularity of surfaces with a linear pencil
Let S be a smooth complex projective surface. Consider a nontrivial fibration \(f: S\to P^ 1.\) Suppose that F, a general fiber of f, is a smooth curve of genus g. The author shows that \(2q-1\leq g\) where \(q=h\) \(1(\omega_ S)\). Furthermore, he gives examples that this inequality is the best possible. The author first reduces the proof to the case when the Albanese map of S is generically finite to one. Then he observes that \(\dim (Im(H^ 0(\omega_ S)\to H^ 0(\omega_ F)))\leq g-q.\) Let A be a general 1- form. The author shows that the rank of the composition map \(H^ 0(\Omega^ 1_ S)^{\wedge}\to^{A}H^ 0(\omega_ S)\to H^ 0(\omega_ F)\) is \(q-1\). Hence \(2q-1\leq g\).
scientific article; zbMATH DE number 4061395