Fibers of cyclic covering fibrations of a ruled surface (Q2286710)

In the paper under review, the author provides a classification result on singular fibers of finite cyclic covering fibrations of a ruled surfaces with use of singularity diagrams. Let \(f: S \rightarrow B\) be a surjective morphism from a smooth complex projective surface \(S\) to a smooth projective curve \(B\) with connected fibers. The datum \((S,f,B)\) or simply \(f\) is called a fibrations. Here the author focuses on primitive cyclic covering fibrations of type \((g,h,n)\), i.e., it is a fibration of genus \(g\) obtained as the relatively minimal model of an \(n\) sheeted cyclic branched covering of another fibration of genus \(h\). In example, hyperelliptic fibrations are fibrations of type \((g,0,2)\). The main result of the paper provides a complete classification of all fibers of \(n=3\) cyclic covering fibrations of genus \(g=4\) of a ruled surface (\(h=0\)). Moreover, the author shows that the signature of a complex surface with this fibration is non-positive by computing the local signature for any fiber. Another result classifies all fibers of hyperelliptic fibrations of genus \(g=3\) into \(12\) types according to the Horikawa index, and furthermore the author shows that finite cyclic covering fibrations of a ruled surface have no multiple fibers if the degree of the covering is greater than \(3\).