Néron’s pairing and relative algebraic equivalence

Abstract

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. Neron 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, Neron gave an expression of his pairing involving intersection multiplicities on the Neron model A of AK over R. When XK is a curve, Gross and Hriljac gave independently an analogous description of Neron’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 Neron 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 Neron 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.