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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Lefschetz pencils and divisors in moduli space

Ivan Smith

Geometry & Topology 5 (2001) 579–608

Erratum: Geometry & Topology 18:1 (2014) 615–616

arXiv: math.SG/0011221

Abstract

We study Lefschetz pencils on symplectic four-manifolds via the associated spheres in the moduli spaces of curves, and in particular their intersections with certain natural divisors. An invariant defined from such intersection numbers can distinguish manifolds with torsion first Chern class. We prove that pencils of large degree always give spheres which behave ‘homologically’ like rational curves; contrastingly, we give the first constructive example of a symplectic non-holomorphic Lefschetz pencil. We also prove that only finitely many values of signature or Euler characteristic are realised by manifolds admitting Lefschetz pencils of genus two curves.

Keywords
Lefschetz pencil, Lefschetz fibration, symplectic four-manifold, moduli space of curves
Mathematical Subject Classification 2000
Primary: 53C15
Secondary: 57R55
References
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Publication
Received: 7 January 2000
Revised: 13 June 2000
Accepted: 4 June 2001
Published: 18 June 2001
Proposed: Gang Tian
Seconded: Ronald Stern, Ronald Fintushel
Authors
Ivan Smith
New College
University of Oxford
OX1 3BN
United Kingdom