Branched covers and pencils on hyperelliptic Lefschetz fibrations

TL;DR

This paper constructs infinite families of symplectic 4-manifolds with hyperelliptic Lefschetz fibrations, showing they are diffeomorphic to complex surfaces formed by fiber sums, and introduces explicit Lefschetz pencils distinct from degree doubling methods.

Contribution

It provides explicit monodromy factorizations for Lefschetz pencils on hyperelliptic fibrations and shows their diffeomorphism to fiber sums of standard complex surfaces, expanding understanding of Lefschetz fibration structures.

Findings

Constructed infinite families of symplectic 4-manifolds with Lefschetz pencils.

Proved these manifolds are diffeomorphic to fiber sums of hyperelliptic Lefschetz fibrations.

Identified explicit Lefschetz pencils different from degree doubling families.

Abstract

Generalizing work of I. Baykur, K. Hayano, and N. Monden (arXiv:1903.02906), we construct infinite families of symplectic 4-dimensional manifolds, obtained as total spaces of Lefschetz pencils constructed by explicit monodromy factorizations. Then, generalizing work of the author (arXiv:2108.04868), we show that each of these manifolds is diffeomorphic to a complex surface that is a fiber sum formed from two standard examples of hyperelliptic Lefschetz fibrations. Consequently, we see that these hyperelliptic Lefschetz fibrations, as well as all fiber sums of them, admit an infinite family of explicitly described Lefschetz pencils, which we observe are different from families formed by the degree doubling procedure.