Geometry & Topology

Lefschetz pencils and divisors in moduli space

Ivan Smith

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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.

Article information

Geom. Topol., Volume 5, Number 2 (2001), 579-608.

Received: 7 January 2000
Revised: 13 June 2000
Accepted: 4 June 2001
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.)
Secondary: 57R55: Differentiable structures

Lefschetz pencil Lefschetz fibration symplectic four-manifold moduli space of curves


Smith, Ivan. Lefschetz pencils and divisors in moduli space. Geom. Topol. 5 (2001), no. 2, 579--608. doi:10.2140/gt.2001.5.579.

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