Inequalities between the Chern numbers of a singular fiber in a family of algebraic curves (Q2846968)

Given a family of curves, the second author defined in [Math. Z. 222, No. 4, 655--676 (1996; Zbl 0864.14016)] and [Acta Math. Vietnam. 35, No. 1, 159--172 (2010; Zbl 1195.14013)] the Chern numbers of a singular fibre. These numbers are essentially the local contribution of the fibres to the global Chern numbers of the total space. In this paper the authors pursue through the attempt to classify singular fibres via these numbers. Some inequalities are given. Moreover, the dual fibre \(F^*\) of a fibre \(F\) is defined; this is a natural generalization of Kodaira's dual fibres for elliptic fibrations [Ann. Math. (2) 78, 1--40 (1963; Zbl 0171.19601)]. The authors prove a duality theorem for these objects. As an application, they classify singular fibres with large or small Chern numbers.