Minimal multiplicity of fiber components in abelian fibrations

TL;DR

This paper proves that in an abelian fibration, the minimal multiplicity among fiber components equals their greatest common divisor, revealing a fundamental property of fiber structure.

Contribution

It establishes a precise relationship between the minimal component multiplicity and their gcd in abelian fibrations, a new insight into fiber component multiplicities.

Findings

Minimal multiplicity equals gcd of component multiplicities

Provides a fundamental property of fiber component structure

Enhances understanding of abelian fibration fibers

Abstract

An abelian fibration is a proper projective surjective map of complex varieties with general fiber an abelian variety. Consider a multiple fiber of an abelian fibration, and let $m_1, ..., m_k$ be the multiplicities of its irreducible components. We prove that the minimum of $m_i$ is equal to their greatest common divisor $gcd(m_1, ..., m_k)$