We show that any pseudo-effective divisor on a normal surface decomposes uniquely into its "integral positive" part and "integral negative" part, which is an integral analog of Zariski decompositions. As an application, we give a generalization of the Kawamata-Viehweg vanishing, Ramanujam's 1-connected vanishing and Miyaoka's vanishing theorems on surfaces. By using this vanishing result, we give a simple proof of Reider-type theorems including the log surface case and the relative case.

Abstract. We study slopes of finite cyclic covering fibrations of a fibered surface. We give the best possible lower bound of the slope of these fibrations. We also give the slope equality of finite cyclic covering fibrations of a ruled surface and observe the local concentration of the global signature of these surfaces on a finite number of fiber germs. We also give an upper bound of the slope of finite cyclic covering fibrations of a ruled surface.

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We give a slope equality for fibered surfaces whose general fiber is a smooth plane curve. As a corollary, we prove a "strong" Durfee-type inequality for isolated hypersurface surface singularities, which implies Durfee's strong conjecture for such singularities with non-negative topological Euler number of the exceptional set of the minimal resolution.2010 Mathematics Subject Classification. 14D06.