mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a Lefschetz pencil is a construction in

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

considered by Solomon Lefschetz, 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 solution set, set of solutions of a system of polynomial equations over the real number, ...

V

Description

A ''pencil'' is a particular kind of linear system of divisors on

V

, namely a one-parameter family, parametrised by the

projective line In projective geometry and mathematics more generally, 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 ...

. This means that in the case of a complex algebraic variety

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 ma ...

over the

Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a Mathematical model, model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents ...

; 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 is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in \mathbb^n of some finite family of homogeneous polynomials that generate a prime ideal, th ...

, 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'

L'=0

for linear forms

L

and

L'

, and the general hyperplane section is

V

intersected with :

\lambda L + \mu L^\prime = 0.\

Then the intersection

J

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 ...

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 component, irreducible ...

V \rightarrow P^1\ :\ p\mapsto \left L'(p):-L(p) \right /math>

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 the space of all directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the poin ...

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 (mathematics), homology cycles of a smooth fiber in a family which vanish in the singular fiber. For example, in a map ...

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. 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 differenti ...

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 may ...

s. It has also been shown that Lefschetz pencils exist in characteristic p 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 function, smooth (differentiable) four-dimensional manifolds, Donaldson–Thomas theory, and his contributions to Kähl ...

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

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 , url-status=dead , title=The topology of symplectic manifolds , journal= Turkish Journal of Mathematics , volume=25 , year=2001 , pages=43–59 , mr=1829078 , archive-url=https://web.archive.org/web/20220206081739/https://journals.tubitak.gov.tr/math/issues/mat-01-25-1/mat-25-1-2-0103-2.pdf , archive-date=2022-02-06 Geometry of divisors

Description

See also

References

Notes

External links