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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''. 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.
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| 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''′, 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
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| :<math>\lambda L + \mu L^\prime = 0.\ </math> | |
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| Then the intersection ''J'' of ''H'' with ''H''′ has [[codimension]] two. There is a [[rational mapping]]
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| :<math>V \rightarrow P^1\ </math>
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| 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''.
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| 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>
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| 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.
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| [[Simon Donaldson]] has found a role for Lefschetz pencils in [[symplectic topology]], leading to more recent research interest in them.
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| ==See also==
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| *[[Picard–Lefschetz theory]]
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| ==References==
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| *S. K. Donaldson, ''Lefschetz Fibrations in Symplectic Geometry'', Doc. Math. J. DMV Extra Volume ICM II (1998), 309-314
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| * {{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 }}
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| ==Notes==
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| {{Reflist}}
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| ==External links==
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| *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).
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| *Gompf, Robert; [http://www.ma.utexas.edu/users/combs/Gompf/gompf00.pdf The Topology of Symplectic Manifolds] (PDF) pp.10-12.
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| [[Category:Geometry of divisors]]
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