# Lefschetz pencil

In mathematics, a **Lefschetz pencil** is a construction in algebraic geometry considered by Solomon Lefschetz, used to analyse the algebraic topology of an algebraic variety *V*. A *pencil* is a particular kind of linear system of divisors on *V*, namely a one-parameter family, parametrised by the projective line. This means that in the case of a complex algebraic variety *V*, a Lefschetz pencil is something like a fibration over the Riemann sphere; but with two qualifications about singularity.

The first point comes up if we assume that *V* is given as a projective variety, and the divisors on *V* are hyperplane sections. Suppose given hyperplanes *H* and *H*′, spanning the pencil — in other words, *H* is given by *L* = 0 and *H*′ by *L*′= 0 for linear forms *L* and *L*′, and the general hyperplane section is *V* intersected with

Then the intersection *J* of *H* with *H*′ has codimension two. There is a rational mapping

which is in fact well-defined only outside the points on the intersection of *J* with *V*. To make a well-defined mapping, some blowing up must be applied to *V*.

The second point is that the fibers may themselves 'degenerate' and acquire singular points (where Bertini's lemma applies, the *general* hyperplane section will be smooth). A Lefschetz pencil restricts the nature of the acquired singularities, so that the topology may be analysed by the vanishing cycle method. The fibres with singularities are required to have a unique quadratic singularity, only.^{[1]}

It has been shown that Lefschetz pencils exist in characteristic zero. They apply in ways similar to, but more complicated than, Morse functions on smooth manifolds.

Simon Donaldson has found a role for Lefschetz pencils in symplectic topology, leading to more recent research interest in them.

## See also

## References

- S. K. Donaldson,
*Lefschetz Fibrations in Symplectic Geometry*, Doc. Math. J. DMV Extra Volume ICM II (1998), 309-314 - {{#invoke:citation/CS1|citation

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## Notes

- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}

## External links

- Gompf, Robert;
*What is a Lefschetz pencil?*; (PDF)*Notices of the American Mathematical Society*; vol. 52, no. 8 (September 2005). - Gompf, Robert; The Topology of Symplectic Manifolds (PDF) pp.10-12.