In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Lefschetz pencil is a construction in
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
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 invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
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. Mo ...
''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 fo ...
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
In algebraic geometry, a complex algebraic variety is an algebraic variety (in the scheme sense or otherwise) over the field of complex number
In mathematics, a complex number is an element of a number system that extends the real numbers ...
''V'', a Lefschetz pencil is something like a
fibration
The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.
Fibrations are used, for example, in postnikov-systems or obstruction theory.
In this article, all map ...
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 pl ...
; but with two qualifications about singularity.
The first point comes up if we assume that ''V'' is given as a
projective variety
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables w ...
, 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 ...
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
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.
For affine and projective algebraic varieties, the codimension equals the ...
two. There is a
rational mapping
In mathematics, in particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties. This article uses the convention that varieties are irreducible.
Definition
Formal de ...
:
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 th ...
must be applied to ''V''.
The second point is that the fibers may themselves 'degenerate' and acquire
singular points (where
Bertini's lemma In mathematics, the theorem of Bertini is an existence and genericity theorem for smooth connected hyperplane sections for smooth projective varieties over algebraically closed fields, introduced by Eugenio Bertini. This is the simplest and broad ...
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 compl ...
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 id ...
. 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 differentiabl ...
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. He i ...
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 H ...
, 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
'' Turkish Journal of Mathematics'' is open-access, peer-reviewed academic journal published electronically and bimonthly by the Scientific and Technological Research Council of Turkey (TÜBITAK). The goal of the journal is to improve the research ...
, volume= 25 , year=2001, pages= 43–59, mr=1829078
Geometry of divisors