Topological uniqueness results for Lefschetz fibrations over the disc

TL;DR

This paper establishes that certain Lefschetz fibrations over the disc are uniquely determined by their singular fibers, providing a classification for fibrations with two I_1 fibers, inspired by mirror symmetry and algebraic geometry.

Contribution

It proves topological uniqueness of Lefschetz fibrations with specific singular fibers and classifies those with two I_1 fibers, connecting to extremal rational elliptic surfaces.

Findings

Lefschetz fibrations are determined by their singular fibers.

Complete classification of fibrations with 2 I_1 fibers.

Connections to mirror symmetry and algebraic geometry.

Abstract

We prove that a Lefschetz fibration over the disc that, after compactification, has the same singular fibers as an extremal rational elliptic surface can be obtained by deleting a singular fiber and a section from the rational extremal elliptic surface, i.e. such a Lefschetz fibration is determined up to topological equivalence by its set of singular fibers. We get a complete clasification of Lefschetz fibrations with 2 $I_1$ fibers as a byproduct of our results. The proof is inspired by homological mirror symmetry and Karpov--Nogin's theorem on constructivity of helices on del Pezzo surfaces. It would be interesting to extend our results to the case of Lefschetz fibrations that, after compactification, have the same singular fibers as an extremal elliptic K3 surface.