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

Let R be a complete discrete valuation ring with algebraically closed residue field k and fraction field K. Let XK be a proper smooth and geometrically connected scheme over K. Néron defined a canonical pairing on XK between 0-cycles of degree zero and divisors which are algebraically equivalent to zero. When XK 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 AK over R. When XK 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 XK.

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

Néron's symbol, Picard functor, Néron models, duality, Grothendieck's pairing
Mathematical Subject Classification 2010
Primary: 14K30
Secondary: 14G40, 14K15, 11G10
Received: 19 February 2011
Revised: 21 December 2011
Accepted: 18 January 2012
Published: 4 December 2012
Cédric Pépin
KU Leuven
Departement Wiskunde
Celestijnenlaan 200B
3001 Heverlee