Infinitely many Lefschetz pencils on 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.