Inequivalent Lefschetz fibrations on rational and ruled surfaces

TL;DR

This paper constructs explicit examples of inequivalent Lefschetz fibrations on rational and ruled surfaces, showing that symplectic 4-manifolds can have multiple distinct Lefschetz structures that are not related by known surgeries.

Contribution

It provides explicit constructions of inequivalent Lefschetz pencils on rational and ruled surfaces, expanding understanding of Lefschetz fibrations on symplectic 4-manifolds.

Findings

Existence of inequivalent Lefschetz fibrations on all rational and ruled surfaces.

Such fibrations cannot be related by fibered Luttinger surgeries.

Every symplectic 4-manifold, after enough blow-ups, admits these inequivalent fibrations.

Abstract

In this short note, we give an explicit construction of inequivalent Lefschetz pencils and fibrations of same genera on blow-ups of all rational and ruled surfaces. This complements our earlier results, concluding that every symplectic 4-manifold, after sufficiently many blow-ups, admits inequivalent Lefschetz pencils and fibrations, which cannot be obtained from one another even via any sequence of fibered Luttinger surgeries.