A degree doubling formula for braid monodromies and Lefschetz pencils

TL;DR

This paper provides an explicit degree doubling formula for braid monodromies and Lefschetz pencils in symplectic 4-manifolds, enhancing understanding of their topological invariants under linear system degree changes.

Contribution

It introduces a novel explicit formula describing how braid monodromy invariants change when the degree of the linear system doubles, impacting the study of symplectic Lefschetz pencils.

Findings

Derived a degree doubling formula for braid monodromies.

Established a similar formula for Lefschetz pencil monodromies.

Enhanced understanding of topological invariants in symplectic geometry.

Abstract

Every compact symplectic 4-manifold can be realized as a branched cover of the complex projective plane branched along a symplectic curve with cusp and node singularities; the covering map is induced by a triple of sections of a "very ample" line bundle. In this paper, we give an explicit formula describing the behavior of the braid monodromy invariants of the branch curve upon degree doubling of the linear system (from a very ample bundle $L^k$ to $L^{2k}$). As a consequence, we derive a similar degree doubling formula for the mapping class group monodromy of Donaldson's symplectic Lefschetz pencils.