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.

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Ivan Smith
New College, Oxford OX1 3BN, UK

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