Multiple fibers of del Pezzo fibrations

TL;DR

This paper proves that terminal three-dimensional del Pezzo fibrations cannot have fibers with multiplicity six or higher, and provides a classification of potential singular point configurations on multiple fibers, along with examples.

Contribution

It establishes a bound on fiber multiplicity in del Pezzo fibrations and offers a classification framework for singularities on multiple fibers.

Findings

Fibers of multiplicity ≥ 6 do not occur in terminal three-dimensional del Pezzo fibrations.

A rough classification of singular point configurations on multiple fibers is provided.

Examples illustrating the possible configurations are given.

Abstract

We prove that a terminal three-dimensional del Pezzo fibration has no fibers of multiplicity $\ge 6$. We also obtain a rough classification possible configurations of singular points on multiple fibers and give some examples.