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PENCILS OF CURVES ON SMOOTH SURFACES

Published online by Cambridge University Press:  23 August 2001

A. MELLE-HERNÁNDEZ
Affiliation:
Departamento de Algebra, Facultad de Matematicas, Universidad Complutense, Ciudad Universitaria sn, Madrid 28040, Spainamelle@eucmos.sim.ucm.es
C. T. C. WALL
Affiliation:
Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 3BX ctcw@liv.ac.uk
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Abstract

Although the theory of singularities of curves - resolution, classification, numerical invariants - goes through with comparatively little change in finite characteristic, pencils of curves are more difficult. Bertini's theorem only holds in a much weaker form, and it is convenient to restrict to pencils such that, when all base points are resolved, the general member of the pencil becomes non-singular. Even here, the usual rule for calculating the Euler characteristic of the resolved surface has to be modified by a term measuring wild ramification.

We begin by describing this background, then proceed to discuss the exceptional members of a pencil. In characteristic 0 it was shown by Há and Lê and by Lê and Weber, using topological reasoning, that exceptional members can be characterised by their Euler characteristics. We present a combinatorial argument giving a corresponding result in characteristic p. We first treat pencils with no base points, and then reduce the remaining case to this.

2000 Mathematical Subject Classification: primary 14H20; secondary 14D05, 14E22, 14F20.

Type
Research Article
Copyright
2001 London Mathematical Society

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