Lefschetz pencils and divisors in moduli space

Smith, Ivan¹

¹ New College, University of Oxford, OX1 3BN, United Kingdom

Geometry & topology, Tome 5 (2001) no. 2, pp. 579-608.

Résumé

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.

Détail

DOI : 10.2140/gt.2001.5.579

Keywords: Lefschetz pencil, Lefschetz fibration, symplectic four-manifold, moduli space of curves

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