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.

Keywords

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

Mathematical Subject Classification 2000

Primary: 53C15

Secondary: 57R55

References

Forward citations

Publication

Received: 7 January 2000

Revised: 13 June 2000

Accepted: 4 June 2001

Published: 18 June 2001

Proposed: Gang Tian

Seconded: Ronald Stern, Ronald Fintushel

Correction: 29 January 2014

Authors

Ivan Smith
New College University of Oxford OX1 3BN United Kingdom

Volume 5, issue 2 (2001)

Ivan Smith