Infinitely many Lefschetz pencils on ruled surfaces

TL;DR

The paper demonstrates that ruled surfaces with negative Euler characteristic admit infinitely many inequivalent Lefschetz pencils of fixed genus, using partial conjugation, and constructs compatible symplectic forms, advancing understanding in low-dimensional topology.

Contribution

It establishes the existence of infinitely many inequivalent Lefschetz pencils on certain ruled surfaces and their blow-ups, introducing partial conjugation as a key technique.

Findings

Infinitely many inequivalent Lefschetz pencils exist on ruled surfaces with negative Euler characteristic.

Constructs compatible symplectic forms for these pencils and their blow-ups.

Provides the first example of this phenomenon in the context of ruled surfaces.

Abstract

We show that any ruled surface $X$ with $\chi(X) < 0$ admits infinitely many inequivalent Lefschetz pencils of fixed genus and number of base points. Our proof proceeds by building infinitely many inequivalent Lefschetz fibrations on a blow-up $X \# 4 \overline{\mathbb{CP}^2}$ of $X$ with constant fiber class, via a mechanism known as partial conjugation. Furthermore, there exists a symplectic form on $X$ compatible with all such pencils, and similarly for the fibrations in $X\#4\overline{\mathbb{CP}^2}$. This provides the first example of this phenomenon and makes progress on Problem 4.98 of the K3 list of problems in low-dimensional topology in the case of ruled surfaces.