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

TL;DR

This paper investigates inequalities and bounds for Chern numbers of singular fibers in algebraic curve families, introduces a dual fiber concept, and classifies fibers based on their Chern numbers.

Contribution

It establishes inequalities and bounds for Chern numbers of singular fibers, introduces the dual fiber and proves a duality theorem, and classifies fibers with extreme Chern numbers.

Findings

Derived inequalities between Chern numbers of singular fibers

Established bounds for Chern numbers of singular fibers

Classified singular fibers with notably large or small Chern numbers

Abstract

In a family of curves, the Chern numbers of a singular fiber are the local contributions to the Chern numbers of the total space. We will give some inequalities between the Chern numbers of a singular fiber as well as their lower and upper bounds. We introduce the dual fiber of a singular fiber, and prove a duality theorem. As an application, we will classify singular fibers with large or small Chern numbers.