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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 $V$ .

Description

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

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

Then the intersection $J$ of $H$ with $H'$ ; has codimension two. There is a rational mapping

V\rightarrow P^{1}\ :\ p\mapsto \left[L'(p):-L(p)\right]

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. 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.

References

Donaldson, Simon K. (1998). "Lefschetz fibrations in symplectic geometry". Documenta Mathematica (Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998)). Extra Volume II: 309–314. MR 1648081.
Griffiths, Phillip; Harris, Joe (1994). Principles of Algebraic Geometry. Wiley Classics Library. Wiley Interscience. p. 509. ISBN 0-471-05059-8.

Notes

^ "Monodromy transformation", Encyclopedia of Mathematics, EMS Press, 2001 [1994]

External links

Gompf, Robert (2005). "What is a Lefschetz pencil?" (PDF). Notices of the American Mathematical Society. 52 (8).
Gompf, Robert (2001). "The topology of symplectic manifolds" (PDF). Turkish Journal of Mathematics. 25: 43–59. MR 1829078. Archived from the original (PDF) on 2022-02-06.

[1] "Monodromy transformation", Encyclopedia of Mathematics, EMS Press, 2001 [1994]

[1]

@@ Line 1: / Line 1: @@
-In [[mathematics]], a '''Lefschetz pencil''' is a construction in [[algebraic geometry]] considered by [[Solomon Lefschetz]], in order to analyse the [[algebraic topology]] of an [[algebraic variety]] ''V''. A ''pencil'' here 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.
+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]] <math>V</math>.
+== Description ==
-The first point comes up if we assume that ''V'' is given as a [[projective variety]], and the divisors on ''V'' are [[hyperplane section]]s. Suppose given hyperplanes ''H''  and ''H''&prime;, spanning the pencil &mdash; in other words, ''H'' is given by ''L'' = 0 and ''H''&prime; by ''L''&prime;= 0 for linear forms ''L'' and ''L''&prime;, and the general hyperplane section is ''V'' intersected with
+A ''pencil'' is a particular kind of [[linear system of divisors]] on <math>V</math>, namely a one-parameter family, parametrised by the [[projective line]]. This means that in the case of a [[complex algebraic variety]] <math>V</math>, 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 <math>V</math> is given as a [[projective variety]], and the divisors on <math>V</math> are [[hyperplane section]]s. Suppose given hyperplanes <math>H</math>  and <math>H'</math>, spanning the pencil &mdash; in other words, <math>H</math> is given by <math>L=0</math>  and <math>H'</math> by <math>L'=0</math> for linear forms <math>L</math> and <math>L'</math>, and the general hyperplane section is <math>V</math> intersected with
-:&lambda;''L'' + &mu; ''L''&prime; = 0.
+:<math>\lambda L + \mu L^\prime = 0.\ </math>
-Then the intersection ''J'' of ''H'' with ''H''&prime; has [[codimension]] two. There is a [[rational mapping]]
+Then the intersection <math>J</math> of <math>H</math> with <math>H'</math>; has [[codimension]] two. There is a [[rational mapping]]
-:''V'' &rarr; ''P''<sup>1</sup>
+:<math>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]] must be applied to ''V''.
+which is in fact well-defined only outside the points on the intersection of <math>J</math> with <math>V</math>. To make a well-defined mapping, some [[blowing up]] must be applied to <math>V</math>.
-The second point is that the fibers may themselves 'degenerate' and acquire [[Mathematical singularity|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 singularties are required to have a unique quadratic singularity, only.<ref>{{Springer|id = m/m064700|title = Monodromy transformation}}</ref>
+The second point is that the fibers may themselves 'degenerate' and acquire [[Mathematical singularity|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.<ref>{{Springer|id = m/m064700|title = Monodromy transformation}}</ref>
-It has been shown that Lefschetz pencils exist in [[characteristic zero]]. They apply in ways similar to, but more complicated than, [[Morse function]]s on [[smooth manifold]]s.
+It has been shown that Lefschetz pencils exist in [[characteristic zero]]. They apply in ways similar to, but more complicated than, [[Morse function]]s on [[smooth manifold]]s. 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 general interest in them.
+[[Simon Donaldson]] has found a role for Lefschetz pencils in [[symplectic topology]], leading to more recent research interest in them.
+==See also==
+*[[Picard–Lefschetz theory]]
 ==References==
+*{{cite journal|first=Simon K.|last= Donaldson|authorlink=Simon Donaldson| title=Lefschetz fibrations in symplectic geometry|journal= Documenta Mathematica |volume= Extra Volume II |year=1998|pages= 309-314|mr=1648081|type=Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998)}}
-*S. K. Donaldson, ''Lefschetz Fibrations in Symplectic Geometry'', Doc. Math. J. DMV Extra Volume ICM II (1998), 309-314
-* {{cite book | author=P. Griffiths | authorlink=Phillip Griffiths | coauthors=[[Joe Harris (mathematician)|J. Harris]] | title=Principles of Algebraic Geometry | series=Wiley Classics Library | publisher=Wiley Interscience | year=1994 | isbn=0-471-05059-8 | page=509 }}
+* {{cite book | first1=Phillip|last1= Griffiths | author1-link=Phillip Griffiths |first2=Joe|last2= Harris |authorlink2=Joe Harris (mathematician)  | title=Principles of Algebraic Geometry | series=Wiley Classics Library | publisher=Wiley Interscience | year=1994 | isbn=0-471-05059-8 | page=509 }}
 ==Notes==
@@ Line 25: / Line 32: @@
 ==External links==
-*Gompf, Robert; [http://www.ma.utexas.edu/users/jwilliam/Documents/me/lefschetzpencil.pdf ''What is a Lefschetz pencil?'']; ([[PDF]]) ''Notices of the American Mathematical Society''; vol. 52, no. 8 (September 2005).
+*{{cite journal|last=Gompf|first= Robert|authorlink=Robert Gompf| url=https://www.ams.org/notices/200508/what-is.pdf|title=What is a Lefschetz pencil?|journal=[[Notices of the American Mathematical Society]]| volume= 52|issue= 8 |year= 2005}}
+* {{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}}
-*Gompf, Robert; [http://www.ma.utexas.edu/users/combs/Gompf/gompf00.pdf The Topology of Symplectic Manifolds] (PDF) pp.10-12
 [[Category:Geometry of divisors]]