Geometry & Topology

Lefschetz pencils and divisors in moduli space

Ivan Smith

Full-text: Open access

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.

Article information

Source
Geom. Topol., Volume 5, Number 2 (2001), 579-608.

Dates
Received: 7 January 2000
Revised: 13 June 2000
Accepted: 4 June 2001
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513883038

Digital Object Identifier
doi:10.2140/gt.2001.5.579

Mathematical Reviews number (MathSciNet)
MR1833754

Zentralblatt MATH identifier
1066.57030

Subjects
Primary: 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.)
Secondary: 57R55: Differentiable structures

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

Citation

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


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