Premium
Pencils of Curves on Smooth Surfaces
Author(s) -
MelleHernández A.,
Wall C. T. C.
Publication year - 2001
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/83.2.257
Subject(s) - pencil (optics) , mathematics , gravitational singularity , euler characteristic , base change , base (topology) , euler's formula , pure mathematics , ramification , mathematics subject classification , resolution (logic) , mathematical analysis , computer science , artificial intelligence , mechanical engineering , engineering
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.