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Melle Hernández, Alejandro and Wall, Charles Terence Clegg (2001) Pencils of curves on smooth surfaces. Proceedings of the London Mathematical Society , 83 (2 ). 257278 . ISSN 00246115

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Official URL: http://plms.oxfordjournals.org/
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 nonsingular. 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.
Item Type:  Article 

Uncontrolled Keywords:  Numerical invariants of singularities; Characteristic p; Singularities of curves; Resolution; Bertini’s theorem; Pencil; Euler characteristic; Wild ramification 
Subjects:  Sciences > Mathematics > Algebraic geometry 
ID Code:  13210 
Deposited On:  06 Sep 2011 07:53 
Last Modified:  06 Feb 2014 09:43 
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