N\'eron'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 X_K be a projective smooth and geometrically connected scheme over K. N\'eron 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 by K-rational points, N\'eron gave an expression of his pairing involving intersection multiplicities on the N\'eron model A of A_K over R. When X_K is a curve, Gross and Hriljac gave independantly an analogous description of N\'eron'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. In this article, we show that these intersection computations are valid for an arbitrary scheme X_K as above and arbitrary 0-cyles of degree zero, by using a proper flat normal and semi-factorial model X of X_K over R. When X_K=A_K is an abelian variety, and X is a semi-factorial compactification of its N\'eron model A, these computations can be used to study the algebraic equivalence on X. We then obtain an interpretation of Grothentieck's duality for the N\'eron model A, in terms of the Picard functor of X over R.