On sections of genus two Lefschetz fibrations

TL;DR

This paper discovers new relations in the mapping class group of genus two surfaces that lead to constructing genus two Lefschetz fibrations with multiple sections, revealing that all holomorphic genus 2 fibrations without separating singular fibers have sections.

Contribution

It introduces novel relations in the mapping class group that enable the construction of specific Lefschetz fibrations with multiple sections, advancing understanding of their structure.

Findings

Constructed a genus two Lefschetz fibration with n disjoint sections for n=1,...,8

Showed any holomorphic genus 2 Lefschetz fibration without separating singular fibers admits a section

Derived new relations in the mapping class group of genus two surfaces

Abstract

In this note we find new relations in the mapping class group of a genus two surface with n boundary components for n=1,..., 8 which induce a genus two Lefschetz fibration $CP^2#13CP^2bar \to S^2$ with n disjoint sections. As a consequence, we observe any holomorphic genus 2 Lefschetz fibration without separating singular fibers admits a section.