Unchaining surgery, branched covers, and pencils on elliptic surfaces

Fuller, Terry

doi:10.2140/agt.2023.23.2867

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Unchaining surgery, branched covers, and pencils on elliptic surfaces

Terry Fuller

Algebraic & Geometric Topology 23 (2023) 2867–2893

DOI: 10.2140/agt.2023.23.2867

Abstract

R İnanç Baykur, Kenta Hayano, and Naoyuki Monden used a technique called unchaining to construct a family of simply connected symplectic $4$ –manifolds $X_{g}^{'} (i)$ for all $g \geq 3$ and $0 \leq i \leq g - 1$ (Geom. Topol. 20 (2016) 2335–2395). Among this family, the manifolds $X_{g}^{'} (g - 2)$ are shown to be symplectic Calabi–Yau $4$ –manifolds. They also showed that each $X_{g}^{'} (i) # \bar{{ℂ ℙ}^{[t] 2}}$ admits a pair of inequivalent genus $g$ Lefschetz pencils. We show how to describe every $X_{g}^{'} (i)$ as a $2$ –fold branched cover of a rational surface, and use this to prove that each $X_{g}^{'} (i)$ is diffeomorphic to the elliptic surface $E (g - i)$ . This has several notable consequences: each symplectic Calabi–Yau they construct is diffeomorphic to K3; for each $n \geq 3$ and $g \geq n$ , the elliptic surface $E (n)$ admits a genus $g$ Lefschetz pencil; and for each $n \geq 3$ and $g \geq n$ , the once blown up elliptic surface $E (n) # \bar{{ℂ ℙ}^{[t] 2}}$ admits a pair of inequivalent genus $g$ Lefschetz pencils.

Keywords

symplectic $4$–manifolds, Lefschetz fibration, Lefschetz pencil

Mathematical Subject Classification

Primary: 57K40, 57K43

References

Publication

Received: 23 August 2021

Revised: 30 November 2021

Accepted: 19 December 2021

Published: 7 September 2023

Authors

Terry Fuller
Department of Mathematics California State University, Northridge Northridge, CA United States

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Volume 23, issue 6 (2023)