Modular invariants and singularity indices of hyperelliptic fibrations

TL;DR

This paper explores the relationship between modular invariants and singularity indices in hyperelliptic fibrations, revealing their equivalence in the semistable case and unifying formulas for relative Chern numbers.

Contribution

It establishes that modular invariants for boundary classes are equivalent to singularity indices and connects Xiao's and Cornalba-Harris's formulas for semistable fibrations.

Findings

Modular invariants equal singularity indices for boundary classes.

Xiao's formula for relative Chern numbers matches Cornalba-Harris's in semistable cases.

Provides a unified view of invariants in hyperelliptic fibrations.

Abstract

The modular invariants of a family of semistable curves are the degrees of the corresponding divisors on the image of the moduli map. The singularity indices were introduced by G. Xiao to classify singular fibers of hyperelliptic fibrations and to compute global invariants locally. In the semistable case, we show that the modular invariants corresponding with the boundary classes are just the singularity indices. As an application, we show that the formula of Xiao for relative Chern numbers is the same as that of Cornalba-Harris in the semistable case.