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

Abstract

Chern numbers of a singular fiber in a family of curves are the local contributions of the fiber to the global Chern numbers of the total space. Our first purpose of this paper is to find the best inequalities between the Chern numbers of a singular fiber. Our second purpose is to try to give a new approach to the classification of singular fibers of genus g. We know that when g is big, there are too many singular fibers of genus g to classify completely (see [5], [7], [8], [16]). In order to get the local-global relations between the invariants, one possible way is to classify singular fibers according to their contributions to the global invariants. To explain this approach, we will classify singular fibers with big or small Chern numbers and give some applications. See the survey [2] for the background of the study on the local-global properties for families of curves. A family of curves of genus g over C is a fibration f : X → C whose general fibers F are smooth curves of genus g, where X is a complex smooth projective surface. The family is called semistable if all of the singular fibers are reduced nodal curves. If X = F ×C and f is just the second projection to C, then we call f a trivial family. If all of the smooth fibers of f are isomorphic to each other, equivalently, f becomes trivial under a finite base change C → C, then f is called isotrivial. We always assume that f is relatively minimal, i.e., there is no (−1)-curve in any singular fiber. When g = 1, Kodaira [6] found the global invariants from the singular fibers. The first Chern number c1(X) is always zero, the second Chern number c2(X) is equal to 12χ(OX) by Noether’s formula, and