In
mathematics, a Lefschetz pencil is a construction in
algebraic geometry considered by
Solomon Lefschetz
Solomon Lefschetz (russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear o ...
, used to analyse the
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
of an
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
''V''.
Description
A ''pencil'' is a particular kind of
linear system of divisors
In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family.
These arose first in the f ...
on ''V'', namely a one-parameter family, parametrised by the
projective line
In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
. This means that in the case of a
complex algebraic variety ''V'', a Lefschetz pencil is something like a
fibration over the
Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
; 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 section In mathematics, a hyperplane section of a subset ''X'' of projective space P''n'' is the intersection of ''X'' with some hyperplane ''H''. In other words, we look at the subset ''X'H'' of those elements ''x'' of ''X'' that satisfy the single line ...
s. 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
In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with all the directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the point with the ...
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 In mathematics, vanishing cycles are studied in singularity theory and other parts of algebraic geometry. They are those homology cycles of a smooth fiber in a family which vanish in the singular fiber.
For example, in a map from a connected comp ...
method. The fibres with singularities are required to have a unique quadratic singularity, only.
It has been shown that Lefschetz pencils exist in
characteristic zero
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive ide ...
. They apply in ways similar to, but more complicated than,
Morse function
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiab ...
s on
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s. It has also been shown that Lefschetz pencils exist in
characteristic p
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive ide ...
for the étale topology.
Simon Donaldson
Sir Simon Kirwan Donaldson (born 20 August 1957) is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds, Donaldson–Thomas theory, and his contributions to Kähler geometry. H ...
has found a role for Lefschetz pencils in
symplectic topology
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the Ha ...
, leading to more recent research interest in them.
See also
*
Picard–Lefschetz theory In mathematics, Picard–Lefschetz theory studies the topology of a complex manifold by looking at the critical points of a holomorphic function on the manifold. It was introduced by Émile Picard for complex surfaces in his book , and extended to ...
References
*
*
Notes
External links
*
* {{cite journal, last=Gompf, first= Robert, authorlink=Robert Gompf, url=http://journals.tubitak.gov.tr/math/issues/mat-01-25-1/mat-25-1-2-0103-2.pdf , title=The topology of symplectic manifolds, journal=
Turkish Journal of Mathematics, volume= 25 , year=2001, pages= 43–59, mr=1829078
Geometry of divisors