Néron’s pairing and relative algebraic equivalence

Pépin, Cédric

doi:10.2140/ant.2012.6.1315

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

Néron's pairing and relative algebraic equivalence

Cédric Pépin

Vol. 6 (2012), No. 7, 1315–1348

DOI: 10.2140/ant.2012.6.1315

Abstract

Let $R$ be a complete discrete valuation ring with algebraically closed residue field $k$ and fraction field $K$ . Let $X_{K}$ be a proper smooth and geometrically connected scheme over $K$ . Néron defined a canonical pairing on $X_{K}$ between $0$ -cycles of degree zero and divisors which are algebraically equivalent to zero. When $X_{K}$ is an abelian variety, and if one restricts to those $0$ -cycles supported on $K$ -rational points, Néron gave an expression of his pairing involving intersection multiplicities on the Néron model $A$ of $A_{K}$ over $R$ . When $X_{K}$ is a curve, Gross and Hriljac gave independently an analogous description of Néron’s pairing, but for arbitrary $0$ -cycles of degree zero, by means of intersection theory on a proper flat regular $R$ -model $X$ of $X_{K}$ .

We show that these intersection computations are valid for an arbitrary scheme $X_{K}$ as above and arbitrary $0$ -cycles of degree zero, by using a proper flat normal and semifactorial model $X$ of $X_{K}$ over $R$ . When $X_{K} = A_{K}$ is an abelian variety, and $X = \bar{A}$ is a semifactorial compactification of its Néron model $A$ , these computations can be used to study the relative algebraic equivalence on $\bar{A} ∕ R$ . We then obtain an interpretation of Grothendieck’s duality for the Néron model $A$ , in terms of the Picard functor of $\bar{A}$ over $R$ . Finally, we give an explicit description of Grothendieck’s duality pairing when $A_{K}$ is the Jacobian of a curve of index one.

Keywords

Néron's symbol, Picard functor, Néron models, duality, Grothendieck's pairing

Mathematical Subject Classification 2010

Primary: 14K30

Secondary: 14G40, 14K15, 11G10

Milestones

Received: 19 February 2011

Revised: 21 December 2011

Accepted: 18 January 2012

Published: 4 December 2012

Authors

Cédric Pépin
KU Leuven Departement Wiskunde Celestijnenlaan 200B 3001 Heverlee Belgium
http://https://perswww.kuleuven.be/~u0079577/

Vol. 6, No. 7, 2012