Geometry & Topology
- Geom. Topol.
- Volume 5, Number 2 (2001), 579-608.
Lefschetz pencils and divisors in moduli space
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.
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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Smith, Ivan. Lefschetz pencils and divisors in moduli space. Geom. Topol. 5 (2001), no. 2, 579--608. doi:10.2140/gt.2001.5.579. https://projecteuclid.org/euclid.gt/1513883038