#### Vol. 6, No. 7, 2012

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Néron's pairing and relative algebraic equivalence

### Cédric Pépin

Vol. 6 (2012), No. 7, 1315–1348
##### 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 $X=\phantom{\rule{0.3em}{0ex}}\overline{\phantom{\rule{0.3em}{0ex}}A}$ is a semifactorial compactification of its Néron model $A$, these computations can be used to study the relative algebraic equivalence on $\phantom{\rule{0.3em}{0ex}}\overline{\phantom{\rule{0.3em}{0ex}}A}∕R$. We then obtain an interpretation of Grothendieck’s duality for the Néron model $A$, in terms of the Picard functor of $\phantom{\rule{0.3em}{0ex}}\overline{\phantom{\rule{0.3em}{0ex}}A}$ 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/