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The specialization index of a variety over a discretely valued field


Authors: Lore Kesteloot and Johannes Nicaise
Journal: Proc. Amer. Math. Soc. 145 (2017), 585-599
MSC (2010): Primary 14G05, 14G20
DOI: https://doi.org/10.1090/proc/13266
Published electronically: August 30, 2016
MathSciNet review: 3577863
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Abstract: Let $ X$ be a proper variety over a henselian discretely valued field. An important obstruction to the existence of a rational point on $ X$ is the index, the minimal positive degree of a zero-cycle on $ X$. This paper introduces a new invariant, the specialization index, which is a closer approximation of the existence of a rational point. We provide an explicit formula for the specialization index in terms of an $ snc$-model, and we give examples of curves where the index equals one but the specialization index is different from one, and thus explains the absence of a rational point. Our main result states that the specialization index of a smooth, proper, geometrically connected $ \mathbb{C}((t))$-variety with trivial coherent cohomology is equal to one.


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Additional Information

Lore Kesteloot
Affiliation: Department of Mathematics, KU Leuven, Celestijnenlaan 200B, 3001 Heverlee, Belgium
Email: lore.kesteloot@wis.kuleuven.be

Johannes Nicaise
Affiliation: Department of Mathematics, Imperial College, South Kensington Campus, London SW72AZ, United Kingdom – and – KU Leuven, Department of Mathematics, Celestijnenlaan 200B, 3001 Heverlee, Belgium
Email: j.nicaise@imperial.ac.uk

DOI: https://doi.org/10.1090/proc/13266
Received by editor(s): August 12, 2015
Received by editor(s) in revised form: March 15, 2016, and April 30, 2016
Published electronically: August 30, 2016
Additional Notes: The research of the first author was supported by a Ph.D. fellowship of the Research Foundation of Flanders (FWO)
Communicated by: Romyar T. Sharifi
Article copyright: © Copyright 2016 American Mathematical Society