Lefschetz pencils on a complex projective plane from a topological viewpoint

2025/09/25 by Ju A Lee, Lee, Ju A
Computer Science · Mathematics · #20F12 #57M07 #57R22 #57R55 #FOS: Mathematics #Geometric Topology (math.GT) #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2509.21069

openalex publication_date 2025/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article, we present a differential topological construction of symplectic Lefschetz pencils of genus ((d-1)(d-2))/(2) with d² base points and 3(d-1)² critical points for arbitrary d≥ 4, analogous to the holomorphic Lefschetz pencils of curves of degree d in ℂP². Moreover, for the case d=4, we derive an explicit monodromy factorization of the genus 3 holomorphic Lefschetz pencil on ℂP² based on the braid monodromy technique and prove that it can also be topologically constructed by breeding the monodromy relations of the genus 1 holomorphic Lefschetz pencils.

Citations

• Classification of genus-1 holomorphic Lefschetz pencils
• Sections of the Matsumoto-Cadavid-Korkmaz Lefschetz fibration
• Topology of holomorphic Lefschetz pencils on the four-torus
• Hurwitz equivalence for Lefschetz fibrations and their multisections
• Lantern relations and rational blowdowns
• A degree doubling formula for braid monodromies and Lefschetz pencils
• Monodromy invariants in symplectic topology

Related

• Classification of genus-1 holomorphic Lefschetz pencils
• Topology of holomorphic Lefschetz pencils on the four-torus
• Monodromy invariants in symplectic topology
• Lefschetz pencils, branched covers and symplectic invariants
• Symplectic structures from Lefschetz pencils in high dimensions