Geometry & Topology, Vol. 5 (2001)
Paper no. 19, pages 579--608.
Lefschetz pencils and divisors in moduli space
Ivan Smith
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
AMS subject classification.
Primary: 53C15.
Secondary: 57R55.
DOI: 10.2140/gt.2001.5.579
E-print: arXiv:math.SG/0011221
Submitted to GT on 7 January 2000.
(Revised 13 June 2000.)
Paper accepted 4 June 2001.
Paper published 18 June 2001.
Notes on file formats
Ivan Smith
New College, Oxford OX1 3BN, UK
Email: smithi@maths.ox.ac.uk
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