Lefschetz pencil

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.

Related Topics:
Mathematics - Algebraic geometry - Solomon Lefschetz - Algebraic topology - Algebraic variety - Linear system of divisors - Projective line - Fibration - Riemann sphere

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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 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

Related Topics:
Projective variety - Hyperplane section

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:λL + μ L′ = 0.

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

Related Topics:
Codimension - Rational mapping

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:V → P1

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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 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.

Related Topics:
Singular point - Bertini's lemma - Vanishing cycle

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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 functions on smooth manifolds.

Related Topics:
Characteristic zero - Morse function - Smooth manifold

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Recent interest in Lefschetz pencils has been because of a role found in symplectic topology, by Simon Donaldson.

Related Topics:
Symplectic topology - Simon Donaldson

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