A note on Lefschetz fibrations on algebraic surfaces

TL;DR

This paper investigates whether pencils of hyperplane sections pulled back to algebraic surfaces via finite maps qualify as Lefschetz pencils, contributing to the understanding of Lefschetz fibrations in algebraic geometry.

Contribution

It examines conditions under which pulled-back pencils on algebraic surfaces are Lefschetz pencils, providing insights into the structure of Lefschetz fibrations on complex surfaces.

Findings

Identifies criteria for pencils to be Lefschetz on algebraic surfaces

Provides examples of Lefschetz and non-Lefschetz pencils

Enhances understanding of Lefschetz fibrations in algebraic geometry

Abstract

Let $S$ be a smooth projective complex algebraic surface and $f\, :\, S\, \longrightarrow\, {\mathbb C}{\mathbb P}^2$ a finite map. Consider a pencil of hyperplane sections on ${\mathbb C}{\mathbb P}^2$ and pull it back to $S$. We address the question whether such a pencil is a Lefschetz pencil on $S$.