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Degenerations and limit Frobenius structures in rigid cohomology

Part of: Arithmetic algebraic geometry Computational number theory Families, fibrations

Published online by Cambridge University Press:  01 February 2011

Alan G. B. Lauder
Alan G. B. Lauder*
Affiliation:
Mathematical Institute, University of Oxford, 24–29 St Giles Oxford OX1 3LB, United Kingdom (email: lauder@maths.ox.ac.uk)

Abstract

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We introduce a ‘limiting Frobenius structure’ attached to any degeneration of projective varieties over a finite field of characteristic p which satisfies a p-adic lifting assumption. Our limiting Frobenius structure is shown to be effectively computable in an appropriate sense for a degeneration of projective hypersurfaces. We conjecture that the limiting Frobenius structure relates to the rigid cohomology of a semistable limit of the degeneration through an analogue of the Clemens–Schmidt exact sequence. Our construction is illustrated, and conjecture supported, by a selection of explicit examples.

MSC classification

Secondary: 11G25: Varieties over finite and local fields 11Y99: None of the above, but in this section 14D06: Fibrations, degenerations
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 14 , 2011 , pp. 1 - 33
Copyright
Copyright © London Mathematical Society 2011

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