A pencil is a particular kind of linear system of divisors on
, namely a one-parameter family, parametrised by the projective line.
This means that in the case of a complex algebraic variety
, a Lefschetz pencil is something like a fibration over the Riemann sphere; but with two qualifications about singularity.
There is a rational mapping which is in fact well-defined only outside the points on the intersection of
To make a well-defined mapping, some blowing up must be applied to
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.
[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.
It has also been shown that Lefschetz pencils exist in characteristic p for the étale topology.
Simon Donaldson has found a role for Lefschetz pencils in symplectic topology, leading to more recent research interest in them.